Factorize the following x3-x2+ax+x-a-1.

# Factorize the following ${x}^{3}-{x}^{2}+\mathit{ax}+x-a-1$.

1. A
$\left(x-1\right)\left({x}^{2}+a+1\right)$
2. B
$\left(x+1\right)\left({x}^{2}+a+1\right)$
3. C
$\left(x-1\right)\left({x}^{2}-a+1\right)$
4. D
$\left(x-1\right)\left({x}^{2}+a-1\right)$

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### Solution:

The given polynomial is ${x}^{3}-{x}^{2}+\mathit{ax}+x-a-1$.
Rewriting the expression by grouping the terms,
$\left({x}^{3}-{x}^{2}\right)+\left(\mathit{ax}-a\right)+\left(x-1\right)$
Taking out the common terms,
${x}^{2}\left(x-1\right)+a\left(x-1\right)+1\left(x-1\right)$
Simplify the expression.
$\left(x-1\right)\left({x}^{2}+a+1\right)$.
Thus, factors are $\left(x-1\right)\left({x}^{2}+a+1\right)$.
Therefore, option 1 is  correct.

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