First, you can readily see that unless white comes up with a forcing move, black could easily cause a lot of trouble for white starting with Q-d5+. So white needs to find a way to keep the heat on black first and does so with Rxd8+. This black king pretty much has to recapture with Kxd8. (Obviously, if black plays K-e7, then Qxf8 is checkmate.)
So after black plays Kxd8, white responds with Qxf8+. The best move for black now is Q-e8. (Because if K-d7, white responds with Q-d6+, then after K-e8, white checkmates with R-c8.)
After black plays Q-e8, white plays R-c8+ winning the black queen for the rook. (Because after Kxc8, white plays Qxe8. Of course, if black plays K-d7, then white checkmate’s with Qxe8.)
I hope you found the answer. Look for another new chess puzzle in my next post!