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3x^{2}\left(6x+1\right)-\left(6x+1\right)
Do the grouping 18x^{3}+3x^{2}-6x-1=\left(18x^{3}+3x^{2}\right)+\left(-6x-1\right), and factor out 3x^{2} in the first and -1 in the second group.
\left(6x+1\right)\left(3x^{2}-1\right)
Factor out common term 6x+1 by using distributive property. Polynomial 3x^{2}-1 is not factored since it does not have any rational roots.