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x^{2}\left(4x+3\right)-11\left(4x+3\right)
Do the grouping 4x^{3}+3x^{2}-44x-33=\left(4x^{3}+3x^{2}\right)+\left(-44x-33\right), and factor out x^{2} in the first and -11 in the second group.
\left(4x+3\right)\left(x^{2}-11\right)
Factor out common term 4x+3 by using distributive property. Polynomial x^{2}-11 is not factored since it does not have any rational roots.