Alternate move

In the Chess Mentor excercise Space Advantage vs. Bishop Pair, Black just moved 9...h5 threatening my rook. Why can't I move 10. Qh3 and threaten mate if he takes my rook on g4? Instead they want me to fight on for 6 more moves instead of 2.

http://www.chess.com/chessmentor/view_lesson.html?id=546

The correct move according to them is Rg3 because according to them: 10.Rg3 is the best square to prepare the doubling of the rooks. Now the f4-pawn is poisoned.

I would gladly give up a rook if it meant checkmate, unless I'm missing something. Granted he doesn't have to take my rook then I could just double them next move.

Yeah, it lets black put his rook on a better defensive square with tempo. However it should explain the move in the course, because there it just says "What about your attacked rook?"