How to solve recurrence $T(n) = 5T(\frac{n}{2}) + n^2\lg^2 n$

I have tried solving it using substitution. Apparently, it is exact for some $ n$ and the order of the general solution can be found from this exact solution.

By substitution I got the following (not sure if it is correct):

$ $ T(n) = 5^kT(n) + \sum_{i = 0}^{k}{5^{i}\left(\frac{n}{2^{i}}\right)^{2}\lg^{2}\left(\frac{n}{2^{i}}\right)}$ $

I am not sure how to proceed from this. I don’t even know if this approach is correct so far. How do I solve this recurrence?