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Simplify the following expression:

[tex]\[
(x+3) - \left[(x+2)\left(x^3-1\right)\right]
\][/tex]

A. [tex]\(-x^4 - 2x^3 + 1\)[/tex]

B. [tex]\(-x^4 - 2x^2 + 2x + 5\)[/tex]

C. [tex]\(-x^4 - 2x^3 + 2x + 5\)[/tex]

D. [tex]\(-x^4 - 2x^3 - x + 2\)[/tex]

Sagot :

GinnyAnswer
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]
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