Originally posted by Mephisto2Your solution is correct.
1.Rxh4 threatens Rxh8#. If black tries to avoid this with Rg8, Rf8 or Re8 then 2.Rh7 with uncoverable mate in 1. Of course, a rook move (other than Rxh4) leads to Qd8#. So, only move:
1. ... Rxh4
2.Kg1 now black has no checks, and white threatens Qd8# . Obly move is
2. .... Rh8
3.Ra8+ to deviate the queen (making a cover of c7 impossible)
3. ... Qxa8 forced
4.Bb7 and mate next move 5.Qc7#, except if 4. .... Qb8, then 5.Qe6#.