Factorize:
$x^{3}-x^{2}+a x+x-a-1$
We have:
$x^{3}-x^{2}+a x+x-a-1=\left(x^{3}-x^{2}\right)+(a x-a)+(x-1)$
$=x^{2}(x-1)+a(x-1)+1(x-1)$
$=(x-1)\left(x^{2}+a+1\right)$
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