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9y^{2}\left(y+2\right)+y+2
Do the grouping 9y^{3}+18y^{2}+y+2=\left(9y^{3}+18y^{2}\right)+\left(y+2\right), and factor out 9y^{2} in 9y^{3}+18y^{2}.
\left(y+2\right)\left(9y^{2}+1\right)
Factor out common term y+2 by using distributive property. Polynomial 9y^{2}+1 is not factored since it does not have any rational roots.