Solution

1. Qe7+ Kg8 2. Rxg6+, followed by Qxc5

Or

1. Qe7+ Kh8 2. Qxf8+

Or 

1. Qe7+ Bf7 (or Rf7)  2. Rxg6+ still gets the black queen, Qxc5.

1. Qh7+ Kxh7 2. Rxe8+ Qe4 3. Bxe4# is checkmate

Note 1. Rxe8+? Rxe8+ gives black a tempo since the white king will be in check.

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