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a^{2}\left(5a+7\right)-3\left(5a+7\right)
Do the grouping 5a^{3}+7a^{2}-15a-21=\left(5a^{3}+7a^{2}\right)+\left(-15a-21\right), and factor out a^{2} in the first and -3 in the second group.
\left(5a+7\right)\left(a^{2}-3\right)
Factor out common term 5a+7 by using distributive property. Polynomial a^{2}-3 is not factored since it does not have any rational roots.