Extended Distributive Property: Using variables

To solve this problem, we will expand the expression $(x-3)(x-6)$ using the distributive property, which involves the following steps:

• Step 1: Multiply the first terms of each binomial
  $(x)(x) = x^2$

• Step 2: Multiply the outer terms of the binomials
  $(x)(-6) = -6x$

• Step 3: Multiply the inner terms of the binomials
  $(-3)(x) = -3x$

• Step 4: Multiply the last terms of each binomial
  $(-3)(-6) = 18$

• Step 5: Combine all the products
  $x^2 - 6x - 3x + 18$

• Step 6: Combine like terms
  $-6x - 3x = -9x$, so we have
  $x^2 - 9x + 18$

Therefore, the expanded form of $(x-3)(x-6)$ is $\boxed{x^2 - 9x + 18}$.

Therefore, the solution to the problem is $x^2 - 9x + 18$. This corresponds to choice 1.

To solve the algebraic expression $(2y-3)(y-4)$, we will apply the distributive property, also known as the FOIL method for binomials. This involves multiplying each term in the first binomial by each term in the second binomial.

• Step 1: Multiply the first terms: $2y \times y = 2y^2$.
• Step 2: Multiply the outer terms: $2y \times -4 = -8y$.
• Step 3: Multiply the inner terms: $-3 \times y = -3y$.
• Step 4: Multiply the last terms: $-3 \times -4 = 12$.

Next, we combine all these results: $2y^2 - 8y - 3y + 12$.

Then, we combine the like terms $-8y$ and $-3y$ to get $-11y$.

Therefore, the expanded expression is $2y^2 - 11y + 12$.

This matches choice (3): $2y^2 - 11y + 12$.

Thus, the solution to the problem is $2y^2 - 11y + 12$.

To solve this problem, we'll apply the distributive property to expand the expression $(3x-1)(x+2)$. Below are the steps:

• Step 1: Distribute each term in the first binomial to each term in the second binomial:

$3x(x) + 3x(2) + (-1)(x) + (-1)(2)$

• Step 2: Calculate each term:

$3x^2 + 6x - x - 2$

• Step 3: Combine like terms:

$3x^2 + (6x - x) - 2 = 3x^2 + 5x - 2$

Thus, the expanded expression is $3x^2 + 5x - 2$.

The correct answer choice is $3x^2 + 5x - 2$, corresponding to choice id="4".

To solve the problem $(5x-2)(3+x)$, we will use the distributive property, specifically the FOIL (First, Outer, Inner, Last) method, to expand the expression:

• Step 1: Multiply the First terms: $5x \times 3 = 15x$.
• Step 2: Multiply the Outer terms: $5x \times x = 5x^2$.
• Step 3: Multiply the Inner terms: $-2 \times 3 = -6$.
• Step 4: Multiply the Last terms: $-2 \times x = -2x$.

Now combine all these products together:

$5x^2 + 15x - 2x - 6$

Combine the like terms $15x$ and $-2x$:

$5x^2 + (15x - 2x) - 6 = 5x^2 + 13x - 6$

Thus, the expanded form of the expression is $5x^2 + 13x - 6$.

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

• Step 1: Apply the distributive property using the FOIL method.
• Step 2: Simplify the expression.
• Step 3: Identify the final simplified polynomial expression.

Now, let’s work through each step:

Step 1: Apply the FOIL method:
(First) Multiply the first terms of each binomial: $3a \cdot 2 = 6a$.
(Outer) Multiply the outer terms: $3a \cdot 3a = 9a^2$.
(Inner) Multiply the inner terms: $-4 \cdot 2 = -8$.
(Last) Multiply the last terms of each binomial: $-4 \cdot 3a = -12a$.

Step 2: Combine the results:
Starting with each term from FOIL: $6a + 9a^2 - 8 - 12a$.
Simplify by combining like terms: $9a^2 + (6a - 12a) - 8 = 9a^2 - 6a - 8$.

Step 3: Identify the resulting polynomial expression:
The expression simplifies to $9a^2 - 6a - 8$.

Therefore, the solution to the problem is $9a^2 - 6a - 8$.

To solve this problem, we will expand the expression $(4a-b)(b+3a)$ using the distributive property:

Firstly, use the distributive property to expand:

• Step 1: Distribute $4a$ across both terms in $(b + 3a)$:
  $4a \cdot b = 4ab$ and $4a \cdot 3a = 12a^2$
• Step 2: Distribute $-b$ across both terms in $(b + 3a)$:
  $-b \cdot b = -b^2$ and $-b \cdot 3a = -3ab$

Combine all these terms:

$4ab + 12a^2 - b^2 - 3ab$

Combine like terms:

• The terms $4ab$ and $-3ab$ combine to give $ab$.

Thus, the simplified form of the expression is:

$12a^2 - b^2 + ab - ab = 12a^2 - b^2 - ab$

Therefore, the solution to the problem is $12a^2 - b^2 - ab$, which corresponds to choice 2.

To solve this problem, we'll expand and simplify the expression $(4y + 3)(3x + 2)$ by applying the distributive property. Let's go through the steps:

• Step 1: Use the distributive property on $(4y + 3)(3x + 2)$.
• Step 2: Multiply each term in $(4y + 3)$ with each term in $(3x + 2)$.
• Step 3: Combine like terms if possible.

Now, let's perform these steps in detail:

Step 1: The expression is given as $(4y + 3)(3x + 2)$. We'll expand this by multiplying each component:

$4y \cdot 3x = 12xy$
$4y \cdot 2 = 8y$
$3 \cdot 3x = 9x$
$3 \cdot 2 = 6$

Step 2: Combine all these products to form the expanded expression:

$12xy + 8y + 9x + 6$

Step 3: Verify if we can combine any like terms. In this case, all terms are different, so no combination is possible.

Thus, the simplified result of the expression $(4y+3)(3x+2)$ is: $12xy + 8y + 9x + 6$.

This matches choice 1 from the provided options.

To solve the problem $(xy + 2a) \cdot (x - 2b)$, we will use the distributive property, commonly referred to as the FOIL method for binomials. This involves multiplying each term in the first binomial by each term in the second binomial:

• First: Multiply the first terms of each binomial: $xy \times x = x^2y$.
• Outside: Multiply the outer terms: $xy \times (-2b) = -2xyb$.
• Inside: Multiply the inner terms: $2a \times x = 2ax$.
• Last: Multiply the last terms: $2a \times (-2b) = -4ab$.

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

$x^2y - 2xyb + 2ax - 4ab$

Thus, the correct expression after using the distributive property and simplifying is $x^2y - 2xyb + 2ax - 4ab$.

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, 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 clear to all the terms' preceding sign is - plus, therefore we'll proceed directly to opening the parentheses,

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

$(\textcolor{red}{7x}+\textcolor{blue}{4})(3x+4)\\ \textcolor{red}{7x}\cdot3x+ \textcolor{red}{7x}\cdot4+\textcolor{blue}{4}\cdot 3x +\textcolor{blue}{4}\cdot4\\ 21x^2+28x+12x+16$

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 treat its exponent as zero power since raising any number to the zero power yields the result 1), we'll use the commutative property of addition, additionally we'll arrange (if needed) the expression from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{21x^2}\textcolor{green}{+28x}\textcolor{green}{+12x}+16\\ \textcolor{purple}{21x^2}\textcolor{green}{+40x}+16$

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 expand and solve the expression $(x+y-z) \cdot (2x-y)$, follow these steps:

Step 1: Apply the distributive property to the expression.
We distribute each term in $(x+y-z)$ to each term in $(2x-y)$.

Step 2: Calculate the products:
- First, distribute $x$ to both $2x$ and $-y$:

• $x \cdot 2x = 2x^2$
• $x \cdot (-y) = -xy$

- Next, distribute $y$ to both $2x$ and $-y$:

• $y \cdot 2x = 2xy$
• $y \cdot (-y) = -y^2$

- Finally, distribute $-z$ to both $2x$ and $-y$:

• $-z \cdot 2x = -2xz$
• $-z \cdot (-y) = yz$

Step 3: Combine all the terms from the above calculations:
$2x^2 - xy + 2xy - y^2 - 2xz + yz$.

Step 4: Simplify by combining like terms:
- Combine $-xy$ and $2xy$ to get $xy$.

Therefore, the expanded expression is:
$2x^2 + xy - y^2 - 2xz + yz$.

This corresponds to choice $1$.

Hence, the correct expanded expression is $2x^2 + xy - y^2 - 2xz + yz$.

To solve the expression $(a+b+2c)\cdot(3a-2b)$, we will apply the distributive property.

• Step 1: Distribute each term of the first expression to every term of the second expression.

• Step 2: Compute the resulting products.

• Step 3: Combine like terms.

Let's execute these steps:
Step 1: Distribute:
Distribute $a$: $a \cdot 3a + a \cdot (-2b) = 3a^2 - 2ab$
Distribute $b$: $b \cdot 3a + b \cdot (-2b) = 3ab - 2b^2$
Distribute $2c$: $2c \cdot 3a + 2c \cdot (-2b) = 6ac - 4bc$

Step 2: Add all these products together:
$3a^2 - 2ab + 3ab - 2b^2 + 6ac - 4bc$

Step 3: Combine like terms:
Combine $-2ab + 3ab$ to get $ab$.

Therefore, the simplified expression is:
$3a^2 + ab - 2b^2 + 6ac - 4bc$.

The correct choice is 4.

Thus, the final expanded expression is $3a^2 + ab - 2b^2 + 6ac - 4bc$.

We begin by solving the two exercises inside of the parentheses:

$4a\times7=112$

We then divide each of the sections by 4:

$\frac{4a\times7}{4}=\frac{112}{4}$

In the fraction on the left side we simplify by 4 and in the fraction on the right side we divide by 4:

$a\times7=28$

Remember that:

$a\times7=a7$

Lastly we divide both sections by 7:

$\frac{a7}{7}=\frac{28}{7}$

$a=4$

We begin by solving the addition exercise in the right parenthesis:

$(7x+3)+14=238$

We then multiply each of the terms inside of the parentheses by 14:

$(14\times7x)+(14\times3)=238$

Following this we solve each of the exercises inside of the parentheses:

$98x+42=238$

We move the sections whilst retaining the appropriate sign:

$98x=238-42$

$98x=196$

Finally we divide the two parts by 98:

$\frac{98}{98}x=\frac{196}{98}$

$x=2$

We begin by solving the addition exercise in the right parenthesis:

$(9+17x)\times7=420$

We then multiply each of the terms inside the parentheses by 7:

$(9\times7)+(17x\times7)=420$

We continue by solving each of the exercises inside of the parentheses:

$63+119x=420$

Following this we rearrange the sections whilst maintaining the appropriate sign:

$119x=420-63$

$119x=357$

Finally we divide the two parts by 119:

$\frac{119}{119}x=\frac{357}{119}$

$x=3$

We know that the area of a rectangle is equal to its length multiplied by its width.

We begin by writing an equation with the available data.

$(4x+x^2)\times(3x+8+5x)$

Next we use the distributive property to solve the equation.

$(4x\times3x)+(4x\times8)+(4x\times5x)+(x^2\times3x)+(x^2\times8)+(x^2\times5x)=$

We then solve each of the exercises within the parentheses:

$12x^2+32x+20x^2+3x^3+16x^2+5x^3=$

Finally we add up all the coefficients of X squared and all the coefficients of X cubed and we obtain the following:

$48x^2+8x^3+32x$