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