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