I believe you have a typo - what you meant to write is this: "If (x+iy)^3=2-2i ...", correct?

Write out the cube of (x+iy) and set it equal to 2-2i, and you get

, hence .

Next if x = y, then this becomes , or , which is certainly true.

To find one solution solve . Then since x=y one cube root is x + xi.

To find the remaining quadratic divide the original equation of by the one root you already know: (z-x-xi). You can do this by long division.