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v^{2}\left(4v+3\right)-2\left(4v+3\right)
Do the grouping 4v^{3}+3v^{2}-8v-6=\left(4v^{3}+3v^{2}\right)+\left(-8v-6\right), and factor out v^{2} in the first and -2 in the second group.
\left(4v+3\right)\left(v^{2}-2\right)
Factor out common term 4v+3 by using distributive property. Polynomial v^{2}-2 is not factored since it does not have any rational roots.