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3x^{2}\left(3x+1\right)+3x+1
Do the grouping 9x^{3}+3x^{2}+3x+1=\left(9x^{3}+3x^{2}\right)+\left(3x+1\right), and factor out 3x^{2} in 9x^{3}+3x^{2}.
\left(3x+1\right)\left(3x^{2}+1\right)
Factor out common term 3x+1 by using distributive property. Polynomial 3x^{2}+1 is not factored since it does not have any rational roots.