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x^{2}\left(5x-3\right)+3\left(5x-3\right)
Do the grouping 5x^{3}-3x^{2}+15x-9=\left(5x^{3}-3x^{2}\right)+\left(15x-9\right), and factor out x^{2} in the first and 3 in the second group.
\left(5x-3\right)\left(x^{2}+3\right)
Factor out common term 5x-3 by using distributive property. Polynomial x^{2}+3 is not factored since it does not have any rational roots.