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