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3x^{2}\left(x+2\right)+x+2
Do the grouping 3x^{3}+6x^{2}+x+2=\left(3x^{3}+6x^{2}\right)+\left(x+2\right), and factor out 3x^{2} in 3x^{3}+6x^{2}.
\left(x+2\right)\left(3x^{2}+1\right)
Factor out common term x+2 by using distributive property. Polynomial 3x^{2}+1 is not factored since it does not have any rational roots.