Sure, let's go through the process step by step to identify which expressions are equivalent to [tex]\(-4(x + 2) - 2x + 4\)[/tex].
1. Start with the original expression:
[tex]\[
-4(x + 2) - 2x + 4
\][/tex]
2. Distribute the [tex]\(-4\)[/tex] through the [tex]\((x + 2)\)[/tex]:
[tex]\[
= -4x - 8 - 2x + 4
\][/tex]
3. Combine the like terms [tex]\(-4x\)[/tex] and [tex]\(-2x\)[/tex] and the constants [tex]\(-8\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[
= -4x - 2x - 8 + 4
\][/tex]
[tex]\[
= -6x - 4
\][/tex]
Now that we have simplified the expression to [tex]\(-6x - 4\)[/tex], let's evaluate each given option to see if they are equivalent.
A. [tex]\(-6x - 4\)[/tex]:
[tex]\[
This matches our simplified expression, so it is equivalent.
\][/tex]
B. [tex]\(-4x + 2 - 2x + 4\)[/tex]:
[tex]\[
Simplify this:
\][/tex]
[tex]\[
= -4x - 2x + 2 + 4
\][/tex]
[tex]\[
= -6x + 6
\][/tex]
This does not simplify to [tex]\(-6x - 4\)[/tex], so it is not equivalent.
C. [tex]\(-10x\)[/tex]:
[tex]\[
This is clearly different from \(-6x - 4\), so it is not equivalent.
\][/tex]
D. [tex]\(-4x + (-8) - 2x + 4\)[/tex]:
[tex]\[
Simplify this:
\][/tex]
[tex]\[
= -4x - 8 - 2x + 4
\][/tex]
[tex]\[
= -6x - 4
\][/tex]
This matches the simplified expression, so it is equivalent.
E. [tex]\(-4x - 2x - 8 + 4\)[/tex]:
[tex]\[
Simplify this:
\][/tex]
[tex]\[
= -6x - 8 + 4
\][/tex]
[tex]\[
= -6x - 4
\][/tex]
This also matches the simplified expression, so it is equivalent.
Thus, the expressions that are equivalent to [tex]\(-4(x + 2) - 2x + 4\)[/tex] are:
- A. [tex]\(-6x - 4\)[/tex]
- D. [tex]\(-4x + (-8) - 2x + 4\)[/tex]
- E. [tex]\(-4x - 2x - 8 + 4\)[/tex]
Therefore, the correct answers are 1, 4, 5.
