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5x^{2}\left(6x+7\right)-\left(6x+7\right)
Do the grouping 30x^{3}+35x^{2}-6x-7=\left(30x^{3}+35x^{2}\right)+\left(-6x-7\right), and factor out 5x^{2} in the first and -1 in the second group.
\left(6x+7\right)\left(5x^{2}-1\right)
Factor out common term 6x+7 by using distributive property. Polynomial 5x^{2}-1 is not factored since it does not have any rational roots.