Answer
$(x+y)(x-y)^{3}$
Work Step by Step
$x^{4}-y^{4}-2x^{3}y+2xy^{3}$
We can write it as
$=(x^{2})^{2}-(y^{2})^{2}-2xy(x^{2}-y^{2})$
$=((x^{2})^{2}-(y^{2})^{2})-2xy(x^{2}-y^{2})$
Factor the difference of squares. The factors are the sum and difference of the expressions being squared.
Using the formula $[(a^{2}-b^{2}) = (a+b)(a-b)]$
$=(x^{2}+y^{2})(x^{2}-y^{2})-2xy(x^{2}-y^{2})$
Factor out common factor $(x^{2}-y^{2})$
$=(x^{2}-y^{2})((x^{2}+y^{2})-2xy))$
$=(x^{2}-y^{2})(x^{2}+y^{2}-2xy)$
$=(x^{2}-y^{2})(x-y)^{2}$
The formula for it can be used here is
$ A^{2}- 2.A.B+B^{2} =(A-B)^{2}$
Again first term is the difference of squares. The factors are the sum and difference of the expressions being squared.
Using the formula $[(a^{2}-b^{2}) = (a+b)(a-b)]$
$=(x+y)(x-y)(x-y)^{2}$
$=(x+y)(x-y)^{3}$ $[ a^{m}\times a^{n} = a^{m+n}]$
$x^{4}-y^{4}-2x^{3}y+2xy^{3}=(x+y)(x-y)^{3}$
