Extended Distributive Property: Is the equality correct?

We solve the right side of the equation using the extended distributive property:$(a+b)\times(c+d)=ac+ad+bc+bd$

$(a+4)(a-5)=a^2-5a+4a-20$

$a^2-a-20$

That is, answer D is the correct one.

To solve this problem, let's expand and simplify the expression on the left-hand side:

$(b-3)(b+7)$

Applying the distributive property (FOIL), we have:

• First: $b \times b = b^2$
• Outer: $b \times 7 = 7b$
• Inner: $-3 \times b = -3b$
• Last: $-3 \times 7 = -21$

Combine these terms:

$b^2 + 7b - 3b - 21$

Simplify by combining like terms:

$b^2 + (7b - 3b) - 21 = b^2 + 4b - 21$

The expression on the left simplifies to $b^2 + 4b - 21$. This is identical to the expression on the right-hand side of the equality.

Since both sides of the equation are equal, the given equality is correct.

Thus, the correct answer is Yes.

To solve this problem, let's scrutinize the expression $(14+a)(a-2)$ by expanding it:

• Distribute each term in the first parenthesis with each term in the second parenthesis:
• $(14+a)(a-2) = 14 \cdot a + 14 \cdot (-2) + a \cdot a + a \cdot (-2)$
• Simplify each part: $14a - 28 + a^2 - 2a$
• Combine like terms: $a^2 + 12a - 28$

Now, compare this expanded and simplified expression $a^2 + 12a - 28$ to the given expression on the right-hand side, $-2a^2 + 12a - 28$.

Observe that the coefficient of $a^2$ is $1$ in our expansion but $-2$ in the right-hand side expression.

Therefore, the equality is incorrect due to the differing coefficients of $a^2$ in the expressions.

Hence, the correct choice is: No, due to the coefficient of $a^2$.

To determine if the expressions are equivalent, we need to expand the right-side expression, $(2x + 3)(3x + 4)$, and compare it with $5x^2 + 7x + 7$.

Let's expand the right-side expression:

• First, use the distributive property (or FOIL):
• $(2x + 3)(3x + 4) = 2x \cdot 3x + 2x \cdot 4 + 3 \cdot 3x + 3 \cdot 4$
• This simplifies to $6x^2 + 8x + 9x + 12$.
• Combine like terms: $6x^2 + (8x + 9x) + 12 = 6x^2 + 17x + 12$.

Now, compare the expanded expression $6x^2 + 17x + 12$ to the left side $5x^2 + 7x + 7$:

• The coefficient of $x^2$ is 6 on the right, but 5 on the left.
• The coefficient of $x$ is 17 on the right, but 7 on the left.
• The constant term is 12 on the right, but 7 on the left.

Since all corresponding coefficients differ between the two sides, the expressions are not equivalent.

Therefore, the correct answer is: No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.

To determine if the given equality $(a+b)(c+d) = ab + cd + ac + bd$ is correct, let's expand $(a+b)(c+d)$ using the distributive property:

Step 1: Use the distributive property to expand $(a+b)(c+d)$. We distribute each term in the first parenthesis by each term in the second parenthesis:

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

Step 2: Combine all the terms obtained from the distributive process:

$ac + ad + bc + bd$

Step 3: Compare the expanded form $ac + ad + bc + bd$ with the right-hand side of the given equality $ab + cd + ac + bd$:

The terms do not match, as the expanded form has terms $ad$ and $bc$ instead of $ab$ and $cd$.

Therefore, the correct expanded form is $ac + ad + bc + bd$. Hence, the given equality is not correct.

The correct answer is: No, it must be $ac + ad + bc + bd$.

To solve this problem, we'll confirm whether $(x+8)(2x-3) = 2x^2+13x-24$ is a true equality by expanding and simplifying the left-hand expression.

Step 1: Use the FOIL method to expand $(x+8)(2x-3)$.

• First: Multiply the first terms: $x \cdot 2x = 2x^2$.

• Outer: Multiply the outer terms: $x \cdot (-3) = -3x$.

• Inner: Multiply the inner terms: $8 \cdot 2x = 16x$.

• Last: Multiply the last terms: $8 \cdot (-3) = -24$.

Step 2: Combine these results:

$2x^2 + (-3x) + 16x + (-24) = 2x^2 + 13x - 24$

Step 3: Compare with the right-hand side:

The expanded form is $2x^2 + 13x - 24$, which matches the right side of the original equation.

Therefore, the expression $(x+8)(2x-3)$ correctly simplifies to $2x^2 + 13x - 24$, verifying the equality.

Thus, the correct answer is: Yes.

To determine the correctness of the equation $(y+9)(2y-2)=2y^2-20y+18$, follow these steps:

• Step 1: Expand the left-hand side expression using the distributive property.

Start by applying the distributive property:
$(y + 9)(2y - 2) = y \cdot 2y + y \cdot (-2) + 9 \cdot 2y + 9 \cdot (-2)$

• Step 2: Calculate each term.

$y \cdot 2y = 2y^2$
$y \cdot (-2) = -2y$
$9 \cdot 2y = 18y$
$9 \cdot (-2) = -18$

• Step 3: Combine the terms.

Now, combine the terms:
$2y^2 - 2y + 18y - 18$

• Step 4: Simplify the expression.

This simplifies to:
$2y^2 + 16y - 18$

• Step 5: Compare with the right-hand side.

The expression $2y^2 + 16y - 18$ does not match the right-hand side $2y^2 - 20y + 18$.

Therefore, the original expression is not correct as given. The discrepancy appears to be in the linear terms. If we adjust one factor in the left-hand side, for example changing $y + 9$ to $y - 9$, we would get:

$(y - 9)(2y - 2) = 2y^2 - 20y + 18$, which would match the right side correctly.

Hence, the correct statement is: No, it must be $(y-9)$ instead of $(y+9)$.

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

• Step 1: Expand the expression on the left side using the distributive property

• Step 2: Simplify the expanded expression

• Step 3: Compare the simplified expression with the given expression on the right side

Now, let's work through each step:
Step 1: Expand using the distributive property. We apply the formula: $(a+b)(c+d) = ac + ad + bc + bd$.

For $(-4-x)(7+x)$, distribute each term:
$(-4)(7) + (-4)(x) + (-x)(7) + (-x)(x)$

Step 2: Simplify the terms:
$-28 - 4x - 7x - x^2$
Combine like terms: $-28 - 11x - x^2$

Step 3: Compare this with the given expression $-28 - 11x - x^2$. Both sides of the equation are identical, indicating the expressions are equivalent.

Therefore, the solution to the problem confirms the expressions are equal, and the correct choice is Yes.

To determine if the equation $(x+8)(x-4) = (x-8)(x+4)$ is true, we'll need to expand both sides and compare their forms.

We begin with the left-hand side:

• $(x+8)(x-4)$

• Expanding using the distributive property, we get:

• $x \cdot x + x \cdot (-4) + 8 \cdot x + 8 \cdot (-4)$

• This simplifies to $x^2 - 4x + 8x - 32$

• Further simplification gives $x^2 + 4x - 32$

Now, let's examine the right-hand side:

• $(x-8)(x+4)$

• Expanding using the distributive property, we obtain:

• $x \cdot x + x \cdot 4 - 8 \cdot x - 8 \cdot 4$

• This simplifies to $x^2 + 4x - 8x - 32$

• Further simplification yields $x^2 - 4x - 32$

Next, compare the results:

The left-hand side is $x^2 + 4x - 32$, while the right-hand side is $x^2 - 4x - 32$.

Note that the coefficients of the $x$ terms are different:

• On the left: The coefficient of $x$ is $+4$.

• On the right: The coefficient of $x$ is $-4$.

Since the coefficients of$x$ differ, the two expressions are not equal.

Therefore, the equality $(x+8)(x-4) = (x-8)(x+4)$ is not correct.

The correct answer to this problem is No, the coefficients of $x$ in contrasting expressions.

To address the question about the equality, we will simplify both sides of the given expression:

The left-hand side of the expression is $(a+4)(x+b+c)$.

• Applying the distributive property, expand $(a+4)$ over $(x+b+c)$:

$(a+4)(x+b+c) = a(x+b+c) + 4(x+b+c)$.

Next, further distribute $a$ and $4$ over each term inside the parentheses:

$a(x+b+c) = ax + ab + ac$
$4(x+b+c) = 4x + 4b + 4c$.

So, the expanded form becomes:

$ax + ab + ac + 4x + 4b + 4c$.

Comparing this with the right-hand side, which is $ax + ab + ac + 4$, observe that:

• Both sides have the terms $ax + ab + ac$.
• However, the left-hand side has additional terms $4x + 4b + 4c$ which the right-hand side does not include.
• The constant term $4$ on the right-hand side does not match these additional terms.

Thus, the equality $(a+4)(x+b+c) = ax + ab + ac + 4$ is not correct.

Therefore, the expression is only right if stated differently. Reviewing the choices:

• Choice 2 correctly presents the restructured expression that would validate the equality: $(x+b+c)a+4$.

Hence, the correct response is choice 2: No, it would be true if the expression were $(x+b+c)a+4$.

To determine if the equality $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, we need to expand and simplify the left-hand side to see if it equals the right-hand side.

First, we expand $(3y+x)(4+2x)$ using the distributive property:

• Multiply $3y$ by $4$, giving $12y$.

• Multiply $3y$ by $2x$, giving $6xy$.

• Multiply $x$ by $4$, giving $4x$.

• Multiply $x$ by $2x$, giving $2x^2$.

Combining all these terms, the left-hand side expands to: $12y + 6xy + 4x + 2x^2$.

Notice that this is precisely the same as the right-hand side: $2x^2 + 6xy + 4x + 12y$.

Therefore, the equality $(3y+x)(4+2x) = 2x^2 + 6xy + 4x + 12y$ holds true.

Thus, the solution to the problem is: $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, and the answer choice is Yes.

To determine if the given algebraic expression is correct, we will expand the left-hand side using the distributive property:

The expression is $(-2a + 3b)(4c + 5a)$.

Step-by-step expansion:

• Multiply $-2a$ by $4c$: $-2a \times 4c = -8ac$.
• Multiply $-2a$ by $5a$: $-2a \times 5a = -10a^2$.
• Multiply $3b$ by $4c$: $3b \times 4c = 12bc$.
• Multiply $3b$ by $5a$: $3b \times 5a = 15ab$.

Combine these results: $-8ac - 10a^2 + 12bc + 15ab$.

Now compare this result with the right-hand side of the given expression $8ac + 10a^2 - 12bc - 15ab$.

We can observe that each corresponding term has the opposite sign.

This shows that the original statement is incorrect.

Therefore, the expression is actually the negative of what was given, so:

The expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$.

Thus, the correct choice is:

No, the expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$.

To solve this problem, we'll employ the distributive property to verify the given equality:

Step 1: Expand the left-hand side expression $(4x+3)(8x+5)$:

• Calculate $4x \cdot 8x = 32x^2$

• Calculate $4x \cdot 5 = 20x$

• Calculate $3 \cdot 8x = 24x$

• Calculate $3 \cdot 5 = 15$

Step 2: Combine like terms:

The expanded form is:
$32x^2 + 20x + 24x + 15$

Combine like terms $20x$ and $24x$:

This gives us $32x^2 + (20x + 24x) + 15 = 32x^2 + 44x + 15$.

Step 3: Compare the expanded form with the right-hand side expression:

The expanded form, $32x^2 + 44x + 15$, matches the right-hand side exactly.

Thus, the given equality $(4x+3)(8x+5) = 32x^2 + 44x + 15$ is correct.

The correct answer is Yes.

To solve this problem, we'll perform the distribution as follows:

• Distribute the term $(2x-3)$ across $(-4+y)$.

• Calculate $(2x) \times (-4)$ and $(2x) \times y$.

• Calculate $(-3) \times (-4)$ and $(-3) \times y$.

Let's compute these multiplications:

Let us expand the left side using distribution:
$(2x - 3)(-4 + y)$
= $(2x)(-4) + (2x)(y) + (-3)(-4) + (-3)(y)$
= $-8x + 2xy + 12 - 3y$.

After simplification, the expression becomes:

$-8x + 2xy - 3y + 12$.

Comparing this with the right-hand side of the original equation $-8x + 2xy - 3y + 12$, we observe that both sides are equal.

Therefore, the two sides of the equation are equal, confirming that the given equality is correct.

The final solution is Yes.