Almost. You left off a square and changed some signs:
(9x²-(y+1)²/9) ((9x²-(y+1)²/9)² -x²(y+1)²)
Now we need to clarify your goal. You just said "determine the product"; does that mean to fully expand (removing all parentheses)?
One way to proceed from here is to distribute the first factor, (9x²-(y+1)²/9), over the other, yielding a cube and some other stuff. If you know a formula for the cube of a binomial, you can use that. Otherwise, just keep distributing, using whatever shortcuts you know.