Extended Distributive Property: Applying the formula

Let's simplify the given expression by opening the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition. Therefore we won't forget of course that the sign preceding the term is an inseparable part of it. We will also apply the rules of sign multiplication and thus we can present any expression in parentheses. We'll open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix. Therefore we'll proceed directly to opening the parentheses,

$$Let's begin:

$(\textcolor{red}{x}+\textcolor{blue}{4})(x+3)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot3+\textcolor{blue}{4}\cdot x +\textcolor{blue}{4}\cdot3\\ x^2+3x+4x+12$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, which we define as terms where the variable (or variables each separately), in this case x, have identical exponents .(In the absence of one of the variables from the expression, we'll consider its exponent as zero power given that raising any number to the power of zero yields 1) We'll apply the commutative property of addition, furthermore we'll arrange (if needed) the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{x^2}\textcolor{green}{+3x}\textcolor{green}{+4x}+12\\ \textcolor{purple}{x^2}\textcolor{green}{+7x}+12$In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

Thus the correct answer is C.

Let's simplify the expression by opening the parentheses using the distributive property:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Therefore, the correct answer is (a).

To solve this problem, we'll apply the FOIL method for multiplying binomials:

• First: Multiply the first terms in each binomial: $(2x)(x) = 2x^2$.
• Outer: Multiply the outer terms in the product: $(2x)(3) = 6x$.
• Inner: Multiply the inner terms: $(y)(x) = xy$.
• Last: Multiply the last terms: $(y)(3) = 3y$.

Next, we combine these results to form the expanded expression:

$2x^2 + 6x + xy + 3y$.

Since terms $6x$ and $xy$ are not like terms, they cannot be combined, resulting in the final expression:

$2x^2 + xy + 6x + 3y$.

Upon reviewing the multiple-choice options, the correct answer is the expanded expression, choice 4: $2x^2 + xy + 6x + 3y$.

When we encounter a multiplication exercise of this type, we know that we must use the distributive property.

Step 1: Multiply the first factor of the first parentheses by each of the factors of the second parentheses.

Step 2: Multiply the second factor of the first parentheses by each of the factors of the second parentheses.

Step 3: Group like terms.

a * (c+3) =

a*c + a*3

4  * (c+3) =

4*c + 4*3

ac+3a+4c+12

There are no like terms to simplify here, so this is the solution!

To solve this problem, we'll perform a step-by-step expansion of the expression $(x+13)(y+4)$ using the distributive property:

• Step 1: Multiply the first terms $(x \cdot y) = xy$.
• Step 2: Multiply the outer terms $(x \cdot 4) = 4x$.
• Step 3: Multiply the inner terms $(13 \cdot y) = 13y$.
• Step 4: Multiply the last terms $(13 \cdot 4) = 52$.

After completing these steps, combine the results:

$xy + 4x + 13y + 52$

This is the final expanded form of the expression. By comparing with the given choices, the correct answer is:

$xy + 4x + 13y + 52$

Therefore, the correct choice is option 3.

Let's simplify the given expression, open the parentheses using the expanded distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an inseparable part of it, and we'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(x-8)(x+y)\\ (\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\$Let's begin then with opening the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{(-8)})(x+y)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot y+\textcolor{blue}{(-8)}\cdot x +\textcolor{blue}{(-8)}\cdot y\\ x^2+xy-8x -8y$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Note that in the expression we got in the last stage there are four different terms, this is because there isn't even one pair of terms where the variables (different ones) have the same exponent, additionally the expression is already organized therefore the expression we got is the final and most simplified form:
$\textcolor{purple}{ x^2}\textcolor{green}{+xy}-8x \textcolor{orange}{-8y}\\$We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We therefore concluded that the correct answer is answer A.

Let's simplify the given expression, open the parentheses using the extended distribution law:

$(\textcolor{red}{t}+\textcolor{blue}{k})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{k}c+\textcolor{blue}{k}d$

Note that in the formula template for the above distribution law, we take as a default that the operation between terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an inseparable part of it, and we'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(12-x)(x-3) \\ (\textcolor{red}{12}+\textcolor{blue}{(-x)})(x+(-3))\\$Let's begin then with opening the parentheses:

$(\textcolor{red}{12}+\textcolor{blue}{(-x)})(x+(-3))\\ \textcolor{red}{12}\cdot x+\textcolor{red}{12}\cdot(-3)+\textcolor{blue}{(-x)}\cdot x +\textcolor{blue}{(-x)}\cdot(-3)\\ 12x-36-x^2 +3x$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step, we'll combine like terms, we'll define like terms as terms where the variable (or variables each separately), in this case x, have identical exponents (in the absence of one of the variables from the expression, we'll consider its exponent as zero power, since raising any number to the zero power yields 1), we'll use the commutative property of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{12x}\textcolor{green}{-36}-x^2\textcolor{purple}{+3x}\\ -x^2\textcolor{purple}{+12x+3x}\textcolor{green}{-36}\\ -x^2\textcolor{purple}{+15x}\textcolor{green}{-36}\\$In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We therefore got that the correct answer is answer A (we used the commutative property of addition to verify this).

Let's simplify the given expression, open the parentheses using the extended distribution law:

$(\textcolor{red}{t}+\textcolor{blue}{b})(c+d)=\textcolor{red}{t}c+\textcolor{red}{t}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an inseparable part of it, we will also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms, in this expression as it's clear, all terms have a plus sign prefix, therefore we'll proceed directly to opening the parentheses,

$$Let's begin then with opening the parentheses:

$(\textcolor{red}{a}+\textcolor{blue}{15})(5+a)\\ \textcolor{red}{a}\cdot 5+\textcolor{red}{a}\cdot a+\textcolor{blue}{15}\cdot 5 +\textcolor{blue}{15}\cdot a\\ 5a+a^2+75+15a$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$x^m\cdot x^n=x^{m+n}$

In the next step we'll combine like terms, we'll define like terms as terms where the variable (or variables each separately), in this case a, have identical exponents (in the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is because any number raised to the power of zero equals 1), we'll use the commutative law of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as power of zero):
$\textcolor{purple}{5a}\textcolor{green}{+a^2}+75\textcolor{purple}{+15a}\\ \textcolor{green}{a^2}\textcolor{purple}{+5a+15a}+75\\ \textcolor{green}{a^2}\textcolor{purple}{+20a}+75\\$In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We therefore got that the correct answer is answer B.

To solve this problem, we'll use the distributive property, also known as the FOIL method when dealing with two binomials.

Let's expand the expression $(7+b)(a+9)$:

• First, apply the distributive property by multiplying each term in the first binomial by each term in the second binomial. This means we will have four operations:
• Step 1: Multiply $7$ by $a$. This gives $7a$.
• Step 2: Multiply $7$ by $9$. This gives $63$.
• Step 3: Multiply $b$ by $a$. This gives $ab$.
• Step 4: Multiply $b$ by $9$. This gives $9b$.

After performing these operations, the expression expands to:

$7a + 63 + ab + 9b$

Rearrange the terms in standard form for the final answer, which is:

$ab + 7a + 9b + 63$

Therefore, the solution to the problem is $ab + 7a + 9b + 63$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the structure of the expression
• Step 2: Apply the difference of squares formula
• Step 3: Simplify the expression

Now, let's work through each step:
Step 1: The expression is $(x+y)(x-y)$, which resembles the difference of squares. Step 2: Using the formula for the difference of squares, $(a+b)(a-b) = a^2 - b^2$, we set $a = x$ and $b = y$. Step 3: Applying the formula, we have:

$(x+y)(x-y) = x^2 - y^2$.

Therefore, the solution to the problem is $x^2-y^2$.

To solve this problem, we need to multiply the binomials $(x-6)$ and $(x+2)$ using the distributive property (FOIL method):

• First: Multiply the first terms: $x \cdot x = x^2$
• Outer: Multiply the outer terms: $x \cdot 2 = 2x$
• Inner: Multiply the inner terms: $-6 \cdot x = -6x$
• Last: Multiply the last terms: $-6 \cdot 2 = -12$

Now, we have the terms: $x^2$, $2x$, $-6x$, and $-12$.
We combine the linear terms:

$x^2 + 2x - 6x - 12 = x^2 - 4x - 12$

This is the expanded form of the quadratic expression in standard form.
Therefore, the solution to the problem is $x^2 - 4x - 12$.

To solve this problem, we will use the FOIL method, which stands for First, Outer, Inner, Last. This helps us to systematically expand the product of two binomials:

• Step 1: Multiply the First terms.

The first terms of each binomial are $x$ and $x$. Multiply these to get $x \times x = x^2$.

• Step 2: Multiply the Outer terms.

The outer terms are $x$ and $-4$. Multiply these to get $x \times -4 = -4x$.

• Step 3: Multiply the Inner terms.

The inner terms are $2$ and $x$. Multiply these to get $2 \times x = 2x$.

• Step 4: Multiply the Last terms.

The last terms are $2$ and $-4$. Multiply these to get $2 \times -4 = -8$.

Now, we combine all these results:

$x^2 - 4x + 2x - 8$

Finally, combine like terms:

Combine $-4x$ and $2x$ to get $-2x$.

The expanded form of the expression is therefore:

$x^2 - 2x - 8$

Thus, the solution to the problem is $x^2 - 2x - 8$, which corresponds to choice 1.

Let's simplify the given expression, open the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an inseparable part of it, and we'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(x-6)(x+8)\\ \downarrow\\ \big(\textcolor{red}{x}+\textcolor{blue}{(-6)}\big)(x+8)\\$Let's begin then with opening the parentheses:

$\big(\textcolor{red}{x}+\textcolor{blue}{(-6)}\big)(x+8)\\ \textcolor{red}{x}\cdot x+\textcolor{red}{x}\cdot8+\textcolor{blue}{(-6)}\cdot x+\textcolor{blue}{(-6)}\cdot8\\ x^2+8x-6x-48$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, we'll define like terms as terms where the variable (or variables each separately), in this case x, have identical exponents (in the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is because raising any number to the zero power yields the result 1), we'll use the commutative property of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{x^2}\textcolor{green}{+8x-6x}\textcolor{orange}{-48}\\ \textcolor{purple}{x^2}\textcolor{green}{+2x}\textcolor{orange}{-48}\\$In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We therefore got that the correct answer is answer A.

We begin by opening the parentheses using the extended distributive property to create a long addition exercise:

We then multiply the first term of the left parenthesis by the first term of the right parenthesis.

We multiply the first term of the left parenthesis by the second term of the right parenthesis.

Now we multiply the second term of the left parenthesis by the first term of the left parenthesis.

Finally, we multiply the second term of the left parenthesis by the second term of the right parenthesis.

In the following way:

$(35\times10)+(35\times5)+(4\times10)+(4\times5)=$

We solve each of the exercises within parentheses:

$350+175+40+20=$

We solve the exercise from left to right:

$350+175=525$

$525+40=565$

$565+20=585$

In order to simplify the given expression we must use the expanded distributive law as seen below:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses, using the mentioned distributive law, and then we'll combine like terms if possible. We'll do this whilst taking into account the correct multiplication of signs:

$(-7-4y)(5x+6)=\\ -35x-42-20xy-24y$Therefore, the correct answer is answer A.

We will use the extended distribution law as seen below in order to simplify the given expression:

$(a+b)(c+d)=ac+ad+bc+bd$

We will begin by performing the multiplication between the pairs of parentheses, using the mentioned distribution law. Then we will proceed to combine like terms if possible. We'll do this whilst taking into account the correct multiplication of signs:

$(2x+3)(-5-x)= \\ -10x-2x^2-15-3x=\\ \boxed{-2x^2-13x-15}$

Therefore, the correct answer is answer D.

We will use the expanded distributive law seen below in order to simplify the given expression:

$(a+b)(c+d)=ac+ad+bc+bd$

We will begin by performing the multiplication between the pairs of parentheses, using the mentioned distributive law. We will combine like terms if possible whilst taking into account the correct multiplication of signs:

$(2x-y)(4-3x)= \\ \boxed{8x-6x^2-4y+3xy}$Therefore, the correct answer is answer C.

We will use the expanded distribution law as seen below in order to simplify the given expression:

$(a+b)(c+d)=ac+ad+bc+bd$

First, we'll perform the multiplication between the pairs of parentheses using the distribution law mentioned, and then we will proceed to combine like terms if possible. We'll do this whilst observing the correct multiplication of signs:

$(3b+7a)(-5a+2b)= \\ -15ab+6b^2-35a^2+14ab=\\ \boxed{6b^2-ab-35a^2}$Therefore, the correct answer is answer B.

To solve the exercise $(2x-3)(5x-7)$, we must expand the expression by using the distributive property, commonly referred to as the FOIL method for binomials.

• First, multiply the first terms of each binomial: $2x \cdot 5x = 10x^2$.
• Outside, multiply the outer terms of the binomials: $2x \cdot -7 = -14x$.
• Inside, multiply the inner terms of the binomials: $-3 \cdot 5x = -15x$.
• Last, multiply the last terms of the binomials: $-3 \cdot -7 = 21$.

After performing these operations, the expanded expression is:

$10x^2 - 14x - 15x + 21$.

The next step is to combine the like terms. In this case, the like terms are the linear terms $-14x$ and $-15x$:

$10x^2 - 14x - 15x + 21 = 10x^2 - 29x + 21$.

Thus, after simplifying, the expression is $10x^2 - 29x + 21$.

Therefore, the solution to the expression $(2x-3)(5x-7)$ is $10x^2 - 29x + 21$.

Let's simplify the given expression and open the parentheses using the extended distribution law:

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition.

Therefore the sign preceding the term is an inseparable part of it. We'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first as an expression where addition operation exists between all terms:

$(2x+3)(-5-x)\\ (\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\$Let's begin by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3})((-5)+(-x))\\ \textcolor{red}{2x}\cdot (-5)+\textcolor{red}{2x}\cdot(-x)+\textcolor{blue}{3}\cdot (-5) +\textcolor{blue}{3} \cdot(-x)\\ -10x-2x^2-15-3x$

In calculating the above multiplications, we used the multiplication table and the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms. Like terms are terms where the variable (or variables each separately), in this case x, have identical exponents. In the absence of one of the variables from the expression, we'll consider its exponent as zero power, due to the fact that raising any number to the power of zero yields the result 1) We'll apply the commutative law of addition, additionally we'll arrange the expression from highest to lowest power from left to right (we'll treat the free number as having zero power):
$\textcolor{purple}{-10x}\textcolor{green}{-2x^2}-15\textcolor{purple}{-3x}\\ \textcolor{green}{-2x^2} \textcolor{purple}{-10x}\textcolor{purple}{-3x}-15\\ \textcolor{green}{-2x^2}\textcolor{purple}{-13x}-15$

In the combining of like terms performed above, we highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term remains an inseparable part of it,

We therefore got that the correct answer is answer D.