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8x^{2}\left(x+6\right)-3\left(x+6\right)
Do the grouping 8x^{3}+48x^{2}-3x-18=\left(8x^{3}+48x^{2}\right)+\left(-3x-18\right), and factor out 8x^{2} in the first and -3 in the second group.
\left(x+6\right)\left(8x^{2}-3\right)
Factor out common term x+6 by using distributive property. Polynomial 8x^{2}-3 is not factored since it does not have any rational roots.