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1. Rh8+ Bxh8 2. Bh6+ Ke7 3. Rxf7+ Kd8 4. Bg5+ Kc8 5. Rxd7 Kxd7 6. Qf7+
I would look at Rh8 first:
1. Rh8 Bh8
The only alternative looks like mate to me: [1. …Ke7 2.Qf7 Kd8 3.Bg5 Bf6 4.Bf6 Kc8 5.Re8 Qe8 6.Qe8#]. Continuing:
2. Bh6! Ke7 (Bg7 3.Qg7 Ke7 4.Qf7#)
3. Rf7
I don’t see anything better than this, but maybe I am missing something here. If white tries 3.Qf7, the black king will escape to d8 and c8, so white must go for the material gain by getting black’s queen:
3. …..Kd8
4. Bg5! Qe7
If black tries 4. …Kc8, he will lose the rook at e8 after white plays Qf7+. Continuing:
5. Qd6
I like this better than an immediate Re7 or Be7, but I haven’t looked thoroughly at thos lines.
5. …..Kc8 (or get mated)
6. Re7 Ree7(Rce7 no better)
7. Be7 Rd7
The other defense to the threat of Qd8# is Nc6, but it might lose more material, though I ma not 100% sure since 8.Bf8 seems to make Qh6 an unstoppable threat against the hapless bishop at h8. Continuing:
8. Qh6 Re7 (what else better?)
9. Qh8 and white has a materially decisive edge.
1. Rh8+
1. Rh8+ Bxh8
2. Bh6+ Ke7
3. Rxf7+ Kd8
4. Bg5+!
now if
4. … Kc8
5. Rxd7 Rxd7
5. Qxe8+
1-0
or
4. … R8e7
5. Qg8+ Qe8
6. Rf8
1-0
1.Rh8+ Bxh8
(1…Ke7 2.Rxf7+ Kd8 3.Bg5+ Kc8 4.Rxd7 Rxh8 5.Rxg7 Rxc2+ 6.Kg3 Ne2+ 7.Kg4 Nd4 8.Rg8+ Rxg8 9.Qxg8+ Kd7)
2.Bh6+ Ke7
(2…Bg7 3.Bxg7+ Kg8 4.Bxe5+ Kf8 5.Qg7+ Ke7 6.Rxf7+ Kd8 7.Rxd7+ Rxd7 8.Qf6+ Kc8 9.Bxd4 Rc7 10.c3 Rd7 11.Kg3 Rg8+ 12.Kf2 Re8 13.Ke3 Ree7 14.f4 Re8 15.e5 dxe5 16.Bxe5 Red8)
3.Rxf7+ Kd8
4.Bg5+ Qe7
(4…Re7 5.Rf8+ Qe8 6.Qxe8#)
{4…Kc8 5.Rxd7 Rxd7
(5…Kxd7 6.Qf7+ Re7 7.Qxe7+ Kc6 8.Qe8+ Rd7 9.Qxh8)
6. Qxe8+ Kc7 7.Qxh8 Nxc2}
5.Qxd6+ Kc8
6.Rxe7 Rexe7
7.Bxe7 Rd7
8.Qh6 Rxe7
9.Qxh8+ Kc7
10.c4 Nc6
11.Qg8 Rd7
12.Qe6 Rd6
13.Qf7+ Rd7
14.Qf8 Rd8
15.Qc5 b6
16.Qe3