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