I left a message about a possible issue with this problem. I think there may be an in-between move that's an improvement on the given solution. Could well be I'm mistaken, though, as I haven't had a chance to check my analysis with a computer.
I think the problem with the line you looked at:
2. RxBg6 Qh1+
3. Qf1
is:
3.. Qe4+ (forking white king and the Rook on g6).
It seems bad things are going to happen here for white in any case, letting the analysis run on a bit 2. RxBg6 is only very slightly worse than whites best move (according to the computer) despite losing the exchange (eventually) after that move.
Regards,
Richard.