Doing induction on recurrences correctly
The error is that $c(n-1)^2 + n^2 = cn^2 - 2cn + c +n^2 \leq (c+1)n^2$, so the constant $c$ is in fact no constant, since it grows with every step.
So you get that when $n$ increases by 1, $c$ also does, so you have $O(n) \cdot n^2$, so $O(n^3)$.