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6x^{2}\left(x-8\right)+5\left(x-8\right)
Do the grouping 6x^{3}-48x^{2}+5x-40=\left(6x^{3}-48x^{2}\right)+\left(5x-40\right), and factor out 6x^{2} in the first and 5 in the second group.
\left(x-8\right)\left(6x^{2}+5\right)
Factor out common term x-8 by using distributive property. Polynomial 6x^{2}+5 is not factored since it does not have any rational roots.