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x^{2}\left(x-8\right)+3\left(x-8\right)
Do the grouping x^{3}-8x^{2}+3x-24=\left(x^{3}-8x^{2}\right)+\left(3x-24\right), and factor out x^{2} in the first and 3 in the second group.
\left(x-8\right)\left(x^{2}+3\right)
Factor out common term x-8 by using distributive property. Polynomial x^{2}+3 is not factored since it does not have any rational roots.