Solvable group of order $pqr^2$

Groups of order $r$ are solvable. Groups of order $pqr$ are solvable. So your group is an extension of two solvable groups, so solvable. In a group of order $pqr$, the Sylow $r$-subgroup is normal, and in the quotient of order $pq$, the Sylow $q$-subgroup is normal.

You are left to show that a group of order $pqr$ is solvable, and this can be done exactly the same way you began the $pqr^2$ case.