Here is a method that doesn't involve manipulating the right hand side, which is more I think of what we're looking for:
Artificially create a factor (x - y) and add canceling terms at the end:
x^3 - y^3 = (x - y)( x^2 ) + ( x - y )( y^2 ) + x^2 * y - y^2 * x
With the final two terms...