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x^{2}\left(4x+7\right)-6\left(4x+7\right)
Do the grouping 4x^{3}+7x^{2}-24x-42=\left(4x^{3}+7x^{2}\right)+\left(-24x-42\right), and factor out x^{2} in the first and -6 in the second group.
\left(4x+7\right)\left(x^{2}-6\right)
Factor out common term 4x+7 by using distributive property. Polynomial x^{2}-6 is not factored since it does not have any rational roots.