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x^{3}\left(x+8\right)+2\left(x+8\right)
Do the grouping x^{4}+8x^{3}+2x+16=\left(x^{4}+8x^{3}\right)+\left(2x+16\right), and factor out x^{3} in the first and 2 in the second group.
\left(x+8\right)\left(x^{3}+2\right)
Factor out common term x+8 by using distributive property. Polynomial x^{3}+2 is not factored since it does not have any rational roots.