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x^{2}\left(2x+1\right)-\left(2x+1\right)
Do the grouping 2x^{3}+x^{2}-2x-1=\left(2x^{3}+x^{2}\right)+\left(-2x-1\right), and factor out x^{2} in the first and -1 in the second group.
\left(2x+1\right)\left(x^{2}-1\right)
Factor out common term 2x+1 by using distributive property.
\left(x-1\right)\left(x+1\right)
Consider x^{2}-1. Rewrite x^{2}-1 as x^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).
\left(x-1\right)\left(x+1\right)\left(2x+1\right)
Rewrite the complete factored expression.