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4x^{2}\left(x+4\right)-3\left(x+4\right)
Do the grouping 4x^{3}-12-3x+16x^{2}=\left(4x^{3}+16x^{2}\right)+\left(-3x-12\right), and factor out 4x^{2} in the first and -3 in the second group.
\left(x+4\right)\left(4x^{2}-3\right)
Factor out common term x+4 by using distributive property. Polynomial 4x^{2}-3 is not factored since it does not have any rational roots.