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2y^{2}\left(7y+4\right)+7y+4
Do the grouping 14y^{3}+8y^{2}+7y+4=\left(14y^{3}+8y^{2}\right)+\left(7y+4\right), and factor out 2y^{2} in 14y^{3}+8y^{2}.
\left(7y+4\right)\left(2y^{2}+1\right)
Factor out common term 7y+4 by using distributive property. Polynomial 2y^{2}+1 is not factored since it does not have any rational roots.