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x^{2}\left(2x-5\right)-3\left(2x-5\right)
Do the grouping 2x^{3}-5x^{2}-6x+15=\left(2x^{3}-5x^{2}\right)+\left(-6x+15\right), and factor out x^{2} in the first and -3 in the second group.
\left(2x-5\right)\left(x^{2}-3\right)
Factor out common term 2x-5 by using distributive property. Polynomial x^{2}-3 is not factored since it does not have any rational roots.