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7z^{2}\left(z+2\right)-9\left(z+2\right)
Do the grouping 7z^{3}+14z^{2}-9z-18=\left(7z^{3}+14z^{2}\right)+\left(-9z-18\right), and factor out 7z^{2} in the first and -9 in the second group.
\left(z+2\right)\left(7z^{2}-9\right)
Factor out common term z+2 by using distributive property. Polynomial 7z^{2}-9 is not factored since it does not have any rational roots.