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)##