Let's go through each expression step-by-step to determine which letter corresponds to which expression.
### First Expression:
Given:
[tex]\[
(4x^3 - 4 + 7x) - (2x^3 - x - 8)
\][/tex]
Simplify inside the parentheses:
[tex]\[
= 4x^3 + 7x - 4 - (2x^3 - x - 8)
\][/tex]
Distribute the negative sign:
[tex]\[
= 4x^3 + 7x - 4 - 2x^3 + x + 8
\][/tex]
Combine like terms:
[tex]\[
= (4x^3 - 2x^3) + (7x + x) + (-4 + 8)
\][/tex]
[tex]\[
= 2x^3 + 8x + 4
\][/tex]
The simplified expression is [tex]\(2x^3 + 8x + 4\)[/tex], which corresponds to option [tex]\( B \)[/tex].
### Second Expression:
Given:
[tex]\[
(-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)
\][/tex]
Combine like terms:
[tex]\[
= x^4 + 2x^4 - 3x^2 + x + 4x - 7
\][/tex]
[tex]\[
= (x^4 + 2x^4) + (-3x^2) + (x + 4x) - 7
\][/tex]
[tex]\[
= 3x^4 - 3x^2 + 5x - 7
\][/tex]
The simplified expression is [tex]\(3x^4 - 3x^2 + 5x - 7\)[/tex], which corresponds to option [tex]\( D \)[/tex].
### Third Expression:
Given:
[tex]\[
(x^2 - 2x)(2x + 3)
\][/tex]
Use the distributive property (FOIL method):
[tex]\[
= x^2(2x + 3) - 2x(2x + 3)
\][/tex]
[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]
Combine like terms:
[tex]\[
= 2x^3 + (3x^2 - 4x^2) - 6x
\][/tex]
[tex]\[
= 2x^3 - x^2 - 6x
\][/tex]
The simplified expression is [tex]\(2x^3 - x^2 - 6x\)[/tex], which corresponds to option [tex]\( A \)[/tex].
### Conclusion:
1. The expression [tex]\((4x^3 - 4 + 7x) - (2x^3 - x - 8)\)[/tex] is equivalent to [tex]\(B\)[/tex].
2. The expression [tex]\((-3x^2 + x^4 + x) + (2x^4 - 7 + 4x)\)[/tex] is equivalent to [tex]\(D\)[/tex].
3. The expression [tex]\((x^2 - 2x)(2x + 3)\)[/tex] is equivalent to [tex]\(A\)[/tex].
So the final answer is:
[tex]\[
\boxed{(B, D, A)}
\][/tex]
