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4w^{2}\left(w-3\right)+w-3
Do the grouping 4w^{3}-12w^{2}+w-3=\left(4w^{3}-12w^{2}\right)+\left(w-3\right), and factor out 4w^{2} in 4w^{3}-12w^{2}.
\left(w-3\right)\left(4w^{2}+1\right)
Factor out common term w-3 by using distributive property. Polynomial 4w^{2}+1 is not factored since it does not have any rational roots.