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2x^{2}\left(x+3\right)-\left(x+3\right)
Do the grouping 2x^{3}+6x^{2}-x-3=\left(2x^{3}+6x^{2}\right)+\left(-x-3\right), and factor out 2x^{2} in the first and -1 in the second group.
\left(x+3\right)\left(2x^{2}-1\right)
Factor out common term x+3 by using distributive property. Polynomial 2x^{2}-1 is not factored since it does not have any rational roots.