Let's work through each expression one step at a time to determine which of the given options (A, B, C, or D) matches the given expressions.
### First Expression: \((4x^3 - 4 + 7x) - (2x^3 - x - 8)\)
1. Distribute the negative sign in the second group:
[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]
This matches expression B, which is:
[tex]\[
B. \, 2x^3 + 8x + 4
\][/tex]
So, \((4x^3 - 4 + 7x) - (2x^3 - x - 8)\) is equivalent to expression B.
### Second Expression: \((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\)
1. Combine like terms directly:
[tex]\[
x^4 + 2x^4 - 3x^2 + x + 4x - 7 = 3x^4 + x - 3x^2 - 7
\][/tex]
This matches expression D, which is:
[tex]\[
D. \, 3x^4 - 3x^2 + 5x - 7
\][/tex]
So, \((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\) is equivalent to expression D.
### Third Expression: \((x^2 - 2x)(2x + 3)\)
1. Distribute the terms:
[tex]\[
x^2 \cdot 2x + x^2 \cdot 3 - 2x \cdot 2x - 2x \cdot 3 = 2x^3 + 3x^2 - 4x^2 - 6x = 2x^3 - x^2 - 6x
\][/tex]
This matches expression A, which is:
[tex]\[
A. \, 2x^3 - x^2 - 6x
\][/tex]
So, \((x^2 - 2x) (2x + 3)\) is equivalent to expression A.
### Summary
[tex]\[
\begin{array}{c}
\left(4x^3 - 4 + 7x\right) - \left(2x^3 - x - 8\right) \text{ is equivalent to expression } B \\
\left(-3x^2 + x^4 + x\right) + \left(2x^4 - 7 + 4x\right) \text{ is equivalent to expression } D \\
\left(x^2 - 2x\right) (2x + 3) \text{ is equivalent to expression } A \\
\end{array}
\][/tex]
So the final answers are:
- \((4x^3 - 4 + 7x) - (2x^3 - x - 8)\) is equivalent to expression B.
- \((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\) is equivalent to expression D.
- [tex]\((x^2 - 2x) (2x + 3)\)[/tex] is equivalent to expression A.
