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