Question

select the expressions that are equivalent to $-4(8d + 2)$.
$-4(2 + 8d)$
$8d - 8$
$-8d - 32$
$2(-4d + 8)$

Explanation:

To determine which expressions are equivalent to \(-4(8d + 2)\), we analyze each option:

Step 1: Simplify the original expression

First, expand \(-4(8d + 2)\) using the distributive property (\(a(b + c) = ab + ac\)):
\[

$$\begin{align*} -4(8d + 2) &= -4 \times 8d + (-4) \times 2 \\ &= -32d - 8 \end{align*}$$

\]

Step 2: Analyze each option

• Option 1: \(-4(2 + 8d)\)

By the commutative property of addition (\(a + b = b + a\)), \(8d + 2 = 2 + 8d\). Thus, \(-4(8d + 2) = -4(2 + 8d)\). This is equivalent.

• Option 2: \(8d - 8\)

The simplified form of the original expression is \(-32d - 8\), which is not equal to \(8d - 8\) (coefficients of \(d\) and constants do not match). This is not equivalent.

• Option 3: \(-8d - 32\)

The simplified form of the original expression is \(-32d - 8\), which is not equal to \(-8d - 32\) (coefficients of \(d\) and constants do not match). This is not equivalent.

• Option 4: \(2(-4d + 8)\)

Expand \(2(-4d + 8)\):
\[
2 \times (-4d) + 2 \times 8 = -8d + 16
\]
This is not equal to \(-32d - 8\) (coefficients of \(d\) and constants do not match). This is not equivalent.

Only the first option (\(-4(2 + 8d)\)) is equivalent to \(-4(8d + 2)\).

Answer:

\(\boldsymbol{-4(2 + 8d)}\) (the first option in the list)