To simplify the given expression:
[tex]\[
(x+3)-\left[(x+2)\left(x^3-1\right)\right]
\][/tex]
we will follow these steps:
1. Distribute [tex]\((x+2)\)[/tex] inside the parentheses:
[tex]\[
(x+2)\left(x^3-1\right) = (x+2)x^3 - (x+2) \cdot 1
\][/tex]
[tex]\[
= x^4 + 2x^3 - x - 2
\][/tex]
2. Substitute this back into the original expression:
[tex]\[
(x+3) - \left[ x^4 + 2x^3 - x - 2 \right]
\][/tex]
3. Distribute the negative sign to the terms inside the brackets:
[tex]\[
= (x + 3) - x^4 - 2x^3 + x + 2
\][/tex]
4. Combine like terms:
[tex]\[
= x - x + 3 + 2 - x^4 - 2x^3
\][/tex]
[tex]\[
= -x^4 - 2x^3 + 5
\][/tex]
Thus, the simplified expression is:
[tex]\[
-x^4 - 2x^3 + 5
\][/tex]
Matching this to the provided choices, we see that it corresponds to option C:
[tex]\[
C. -x^4 - 2x^3 + 2x + 5
\][/tex]
