Sure, let's go through the problem step-by-step:
### First Expression
Given Expression: [tex]\((4x^3 - 4 + 7x) - (2x^3 - x - 8)\)[/tex]
1. Distribute the negative sign inside the second parenthesis:
[tex]\[
(4x^3 - 4 + 7x) - 2x^3 + x + 8
\][/tex]
2. Combine like terms:
[tex]\[
(4x^3 - 2x^3) + (7x + x) + (-4 + 8)
\][/tex]
[tex]\[
2x^3 + 8x + 4
\][/tex]
3. The simplified expression matches expression B: [tex]\(2x^3 + 8x + 4\)[/tex].
### Second Expression
Given Expression: [tex]\((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\)[/tex]
1. Combine like terms directly:
[tex]\[
x^4 + 2x^4 - 3x^2 + x + 4x - 7
\][/tex]
[tex]\[
3x^4 - 3x^2 + 5x - 7
\][/tex]
2. The simplified expression matches expression D: [tex]\(3x^4 - 3x^2 + 5x - 7\)[/tex].
### Third Expression
Given Expression: [tex]\((x^2 - 2x)(2x + 3)\)[/tex]
1. Distribute each term in the first parenthesis by each term in the second parenthesis:
[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]
2. Combine like terms:
[tex]\[
2x^3 - x^2 - 6x
\][/tex]
3. The simplified expression matches expression A: [tex]\(2x^3 - x^2 - 6x \)[/tex].
### Final Answers
1. [tex]\((4 x^3 - 4 + 7x) - (2 x^3 - x - 8)\)[/tex] is equivalent to expression B.
2. [tex]\((-3 x^2 + x^4 + x) + (2 x^4 - 7 + 4x)\)[/tex] is equivalent to expression D.
3. [tex]\((x^2 - 2x)(2x + 3)\)[/tex] is equivalent to expression A.
Thus, the answers to the expressions are:
1. B
2. D
3. A
