2011/032) factorize x^4+2x^3y-3x^2y^2-4xy^3-y^4?

seeing coefficient of x^4 as 1 and y^4 is -1 and this is homogeneous in x y

that is power of x and y sum is 4 that is same

we see that if shall be (x^2+axy + y^2)(x^2+bxy-y^2)

= x^2 + x^3y(a+b) + x^2y^2(ab) + (b-a) xy^2 - y^4

so a + b = 2, ab = -3 and a-b = 4

a + b = 2, and a-b = 4 give a = 3 and b = -1 and this satisfies
ab = -3

so we get (x^2+axy + y^2)(x^2+bxy-y^2)
= (x^2+3xy + y^2)(x^2-xy-y^2)