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2y^{2}\left(4y-3\right)-\left(4y-3\right)
Do the grouping 8y^{3}-6y^{2}-4y+3=\left(8y^{3}-6y^{2}\right)+\left(-4y+3\right), and factor out 2y^{2} in the first and -1 in the second group.
\left(4y-3\right)\left(2y^{2}-1\right)
Factor out common term 4y-3 by using distributive property. Polynomial 2y^{2}-1 is not factored since it does not have any rational roots.