Puzzle from Tactics Trainer


white's moves. I did find most of this easy except the last two moves to solve it.
Bx P+, Q b1+, RB7, Qf5, Bishop x kn, Kn g5, Q x Bishop+, Q b6+, RB3 which solves the puzzle, however I think there is an error in the puzzle unless I'm wrong. Why can't blacks queen go Qe1+ forcing whites king to h2. Black queen than takes the pawn preventing the rook from coming over to eventually mate. ?? Does that not seem to work, analysis that for me. Yes black isn't in the best positon still, but hey they may have a fighting chance at the moment it even looks drawish you play my follow up moves.