How can I solve the recurrence $T(n) = 4T(n/2) + n^2\log^2n$? (without master theorem)
You want to use the Master theorem. Writing $T(n)=aT(n/b)+f(n)$, we get that $a=4$, $b=2$, and $f(n)=n^2\log^2n$. This gives $\log_b(a)=2$. This is the same exponent on $f(n)$, but you have an extra $\log^2n$. This fits case 2 on the Wikipedia page. So we get $O(n^2)$ with an extra $\log^3n$. This gives an overall result of $T(n)=O(n^2\log^3n)$.