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a^{2}\left(a+1\right)-3\left(a+1\right)
Do the grouping a^{3}+a^{2}-3a-3=\left(a^{3}+a^{2}\right)+\left(-3a-3\right), and factor out a^{2} in the first and -3 in the second group.
\left(a+1\right)\left(a^{2}-3\right)
Factor out common term a+1 by using distributive property. Polynomial a^{2}-3 is not factored since it does not have any rational roots.