Let’s analyze the given expressions step-by-step and match them with the correct options.
### Step 1: Simplify the first expression
Given expression:
[tex]\[
(4x^3 - 4 + 7x) - (2x^3 - x - 8)
\][/tex]
Simplify it:
1. Distribute the negative sign:
[tex]\[
4x^3 - 4 + 7x - 2x^3 + x + 8
\][/tex]
2. Combine like terms:
[tex]\[
(4x^3 - 2x^3) + (7x + x) + (-4 + 8)
= 2x^3 + 8x + 4
\][/tex]
So, the simplified expression is [tex]\(2x^3 + 8x + 4\)[/tex]. This corresponds to expression B.
### Step 2: Simplify the second expression
Given expression:
[tex]\[
(-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)
\][/tex]
Combine like terms:
[tex]\[
x^4 + 2x^4 - 3x^2 + x + 4x - 7
= 3x^4 - 3x^2 + 5x - 7
\][/tex]
So, the simplified expression is [tex]\(3x^4 - 3x^2 + 5x - 7\)[/tex]. This corresponds to expression D.
### Step 3: Expand and simplify the third expression
Given expression:
[tex]\[
(x^2 - 2x)(2x + 3)
\][/tex]
Distribute:
[tex]\[
x^2 \cdot 2x + x^2 \cdot 3 - 2x \cdot 2x - 2x \cdot 3
\][/tex]
[tex]\[
= 2x^3 + 3x^2 - 4x^2 - 6x
\][/tex]
[tex]\[
= 2x^3 - x^2 - 6x
\][/tex]
So, the expanded and simplified expression is [tex]\(2x^3 - x^2 - 6x\)[/tex]. This corresponds to expression A.
### Final Selections
[tex]\[
(4x^3 - 4 + 7x) - (2x^3 - x - 8) \text{ is equivalent to expression } B
\][/tex]
[tex]\[
(-3x^2 + x^4 + x) + (2x^4 - 7 + 4x) \text{ is equivalent to expression } D
\][/tex]
[tex]\[
(x^2 - 2x)(2x + 3) \text{ is equivalent to expression } A
\][/tex]
Answers:
[tex]\[
B \quad D \quad A
\][/tex]
