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a^{2}\left(b+1\right)-\left(b+1\right)
Do the grouping a^{2}b+a^{2}-b-1=\left(a^{2}b+a^{2}\right)+\left(-b-1\right), and factor out a^{2} in the first and -1 in the second group.
\left(b+1\right)\left(a^{2}-1\right)
Factor out common term b+1 by using distributive property.
\left(a-1\right)\left(a+1\right)
Consider a^{2}-1. Rewrite a^{2}-1 as a^{2}-1^{2}. The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).
\left(a-1\right)\left(a+1\right)\left(b+1\right)
Rewrite the complete factored expression.