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