I haven't been able to spot any logical flaws (doesn't mean there aren't any) but the hint might just be trying to tell you that your proof is unnecessarily complex. If it were just that there are easier methods then I don't think that would be good enough reason to criticise it, but you did quite a lot of work only to create a problem harder than the original question. Near the end, starting from this line:

4x^(2) \+ 8xy + 4y^(2) <= 4xy

you prove indirectly that 4x^(2) \+ 4xy + 4y^(2) <= 0 is contradictory, using the fact proved early on that xy >= 0. But as soon as you had that, you could have proved that 4x^(2) \+ 6xy + 4y^(2) can't be negative using the same method, by considering 4x^(2) \+ 4y^(2) <= -6xy <= 0.