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z^{2}\left(z-6\right)-5\left(z-6\right)
Do the grouping z^{3}-6z^{2}-5z+30=\left(z^{3}-6z^{2}\right)+\left(-5z+30\right), and factor out z^{2} in the first and -5 in the second group.
\left(z-6\right)\left(z^{2}-5\right)
Factor out common term z-6 by using distributive property. Polynomial z^{2}-5 is not factored since it does not have any rational roots.