Solving $T(n)=4T(n/2)+n^2$
You must subtract a lower order term in order to strengthen your inductive hypothesis: try

$T(n) \leq c_1 n^2 \lg n- c_2 n$ and you will be able to show by substitution that $T(n) = O(n^2 \lg n)$ for appropriate values of the constants ($c_2 &gt; 1$ and $c_1$ big enough to correctly handle the initial conditions).