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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)##