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x^{4}\left(x+1\right)+x+1
Do the grouping x^{5}+x^{4}+x+1=\left(x^{5}+x^{4}\right)+\left(x+1\right), and factor out x^{4} in x^{5}+x^{4}.
\left(x+1\right)\left(x^{4}+1\right)
Factor out common term x+1 by using distributive property. Polynomial x^{4}+1 is not factored since it does not have any rational roots.