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x^{2}\left(4x+1\right)-3\left(4x+1\right)
Do the grouping 4x^{3}+x^{2}-12x-3=\left(4x^{3}+x^{2}\right)+\left(-12x-3\right), and factor out x^{2} in the first and -3 in the second group.
\left(4x+1\right)\left(x^{2}-3\right)
Factor out common term 4x+1 by using distributive property. Polynomial x^{2}-3 is not factored since it does not have any rational roots.