Answer
$2x^{2}(x+1)(2x^{3}-3x-4)$
Work Step by Step
$2 $ is a factor of all three terms
$x^{2}$ also, and so is
$(x+1).$
$GCF=2x^{2}(x+1)$
$ \begin{array}{ccc}
4x^{5}(x+1) & -6x^{3}(x+1) & -8x^{2}(x+1)=\\
=2x^{2}(x+1)\cdot 2x^{3} & +2x^{2}(x+1)\cdot(-3x) & +2x^{2}(x+1)\cdot(-4)
\end{array}$
= $2x^{2}(x+1)(2x^{3}-3x-4)$