Answer
$(x^{2}+y^{2})(x+y)(x-y)$
Work Step by Step
Recognize that $x^{4}=(x^{2})^{2},$ and $y^{4}=(y^{2})^{2}.$
$x^{4}-y^{4}= $
...write the difference as $a^{2}-b^{2}$.
$(x^{2})^{2}-(y^{2})^{2} =$
...factor using the rule for a difference of two squares.
($a^{2}-b^{2}=(a+b)(a-b),\quad a=x^{2},\ b=y^{2}$)
$=(x^{2}+y^{2})(x^{2}-y^{2})$
... the first parentheses is a sum of squares. There is no special formula for that. But, the second parentheses contain a difference of two squares...
$=(x^{2}+y^{2})(x+y)(x-y)$
