Solve by factorization

x^4 + x^3 y - x^3 y^2 - x^2 y^3 + x y^2 - y^4

Rearrange the terms to get:
x^4 - x^3y^2 + x^3y - x^2y^3 + xy^2 - y^4
(x^4 - x^3y^2) + (x^3y - x^2y^3) + (xy^2 - y^4)

You can then factor by grouping
x^3(x - y^2) + x^2y(x - y^2) + y^2(x - y^2)
(x - y^2)(x^3 + x^2y + y^2)

x^3(x + y) - x^2y^2(x + y) + y^2(x - y^2)
(x + y)(x^3 - x^2y^2) + y^2(x - y^2)
(x + y)(x^2)(x - y^2) + y^2 (x - y^2)

(x - y^2) [x^2 ( x + y) + y^2)
(x - y^2) [ x^3 + x^2y + y^2]