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x^{2}\left(x-6\right)+4\left(x-6\right)
Do the grouping x^{3}-6x^{2}+4x-24=\left(x^{3}-6x^{2}\right)+\left(4x-24\right), and factor out x^{2} in the first and 4 in the second group.
\left(x-6\right)\left(x^{2}+4\right)
Factor out common term x-6 by using distributive property. Polynomial x^{2}+4 is not factored since it does not have any rational roots.