Answer
$y(y^{2}+1)(y^{4}-y^{2}+1)$
Work Step by Step
$ y^{7}+y\qquad$...factor out the common term, $y$.
$=y(y^{6}+1)\qquad$...recognize a sum of two cubes:$\mathrm{A}^{3}+\mathrm{B}^{3} = (\mathrm{A}+\mathrm{B})(\mathrm{A}^{2} - \mathrm{A}\mathrm{B} + B^{2})$
$=y((y^{2})^{3}+1^{3})$
$=y(y^{2}+1)(y^{4}-y^{2}+1)$
