How do you simplify ##(x y – y ) ^ { 2} – ( x ^ { 2} + y ^ { 2} )##?

##x(xy^2 – 2y^2 – x)##

First you need to square the term by cross multiplying:

##(xy – y)(xy – y) – (x^2 + y^2)##

##(x^2y^2 – xy^2 – xy^2 + y^2) – (x^2 + y^2)##

##(x^2y^2 – 2xy^2 + y^2) – (x^2 + y^2)##

We can now remove the parenthesis from the two terms keeping in mind to make sure we get the signs correct:

##x^2y^2 – 2xy^2 + y^2 – x^2 color(red)(-) y^2##

We can now group like terms:

##x^2y^2 – 2xy^2 – x^2 + y^2 – y^2##

##x^2y^2 – 2xy^2 – x^2 + 0##

##x^2y^2 – 2xy^2 – x^2##

Factoring this gives:

##x(xy^2 – 2y^2 – x)##