Let's break down each expression and compare it to the choices given.
1. For the expression [tex]\((4x^3-4+7x) - (2x^3-x-8)\)[/tex]:
- First, simplify inside the parentheses:
[tex]\[
(4x^3 - 4 + 7x) - (2x^3 - x - 8)
\][/tex]
- Distribute the negative sign:
[tex]\[
= 4x^3 - 4 + 7x - 2x^3 + x + 8
\][/tex]
- Combine like terms:
[tex]\[
= (4x^3 - 2x^3) + (7x + x) + (-4 + 8)
\][/tex]
[tex]\[
= 2x^3 + 8x + 4
\][/tex]
- Comparing this with the given choices:
[tex]\[
2x^3 + 8x + 4 \quad \text{corresponds to B}
\][/tex]
2. For the expression [tex]\((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\)[/tex]:
- Combine like terms within the parentheses:
[tex]\[
= (-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)
\][/tex]
- Combine the expressions:
[tex]\[
= x^4 + 2x^4 - 3x^2 + x + 4x - 7
\][/tex]
- Simplify further:
[tex]\[
= (x^4 + 2x^4) + (-3x^2) + (x + 4x) - 7
\][/tex]
[tex]\[
= 3x^4 - 3x^2 + 5x - 7
\][/tex]
- Comparing this with the given choices:
[tex]\[
3x^4 - 3x^2 + 5x - 7 \quad \text{corresponds to D}
\][/tex]
3. For the expression [tex]\((x^2 - 2x)(2x + 3)\)[/tex]:
- Use the distributive property:
[tex]\[
(x^2 - 2x)(2x + 3)
\][/tex]
- Apply the distributive property:
[tex]\[
= x^2 \cdot 2x + x^2 \cdot 3 - 2x \cdot 2x - 2x \cdot 3
\][/tex]
- Simplify each term:
[tex]\[
= 2x^3 + 3x^2 - 4x^2 - 6x
\][/tex]
- Combine like terms:
[tex]\[
= 2x^3 + (3x^2 - 4x^2) - 6x
\][/tex]
[tex]\[
= 2x^3 - x^2 - 6x
\][/tex]
- Comparing this with the given choices:
[tex]\[
2x^3 - x^2 - 6x \quad \text{corresponds to A}
\][/tex]
So, the expressions correspond to:
1. [tex]\( (4x^3 - 4 + 7x) - (2x^3 - x - 8) \)[/tex] is equivalent to expression B.
2. [tex]\( (-3x^2 + x^4 + x) + (2x^4 - 7 + 4x) \)[/tex] is equivalent to expression D.
3. [tex]\( (x^2 - 2x)(2x + 3) \)[/tex] is equivalent to expression A.
