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2c^{2}\left(4c+7\right)-\left(4c+7\right)
Do the grouping 8c^{3}+14c^{2}-4c-7=\left(8c^{3}+14c^{2}\right)+\left(-4c-7\right), and factor out 2c^{2} in the first and -1 in the second group.
\left(4c+7\right)\left(2c^{2}-1\right)
Factor out common term 4c+7 by using distributive property. Polynomial 2c^{2}-1 is not factored since it does not have any rational roots.