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2z^{2}\left(z+3\right)+3\left(z+3\right)
Do the grouping 2z^{3}+6z^{2}+3z+9=\left(2z^{3}+6z^{2}\right)+\left(3z+9\right), and factor out 2z^{2} in the first and 3 in the second group.
\left(z+3\right)\left(2z^{2}+3\right)
Factor out common term z+3 by using distributive property. Polynomial 2z^{2}+3 is not factored since it does not have any rational roots.