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5b^{3}\left(b-3\right)-\left(b-3\right)
Do the grouping 5b^{4}-15b^{3}+3-b=\left(5b^{4}-15b^{3}\right)+\left(3-b\right), and factor out 5b^{3} in the first and -1 in the second group.
\left(b-3\right)\left(5b^{3}-1\right)
Factor out common term b-3 by using distributive property. Polynomial 5b^{3}-1 is not factored since it does not have any rational roots.