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y^{2}\left(5y+3\right)+2\left(5y+3\right)
Do the grouping 5y^{3}+3y^{2}+10y+6=\left(5y^{3}+3y^{2}\right)+\left(10y+6\right), and factor out y^{2} in the first and 2 in the second group.
\left(5y+3\right)\left(y^{2}+2\right)
Factor out common term 5y+3 by using distributive property. Polynomial y^{2}+2 is not factored since it does not have any rational roots.