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2x\left(xy+z^{2}\right)+y^{2}\left(xy+z^{2}\right)
Do the grouping 2x^{2}y+2xz^{2}+y^{2}z^{2}+xy^{3}=\left(2x^{2}y+2xz^{2}\right)+\left(y^{2}z^{2}+xy^{3}\right), and factor out 2x in the first and y^{2} in the second group.
\left(xy+z^{2}\right)\left(2x+y^{2}\right)
Factor out common term xy+z^{2} by using distributive property.