The immediate threat is the queen taking your pawn directly in front of the king and putting you in checkmate (Qxh3). so you need to prevent this buy either blocking or stopping the move. The obvious move then is to move the rook on the open file straight up to check the king (Re8), preventing the queen move described earlier. the king will then move towards your rook to get out of check and threaten the rook (you just moved) (Kd7). So you have to move it away (or lose it), moving to threaten the queen (and still prevent the queen checkmate) would seem the move to make (Re3). To either force the queen to take your rook (Qxe3) and you get the queen via the knight (Kxe3) or the queen has to move away. the bishop can't take the rook (Bxe3) as it's pinned, because if it did move the king is in check. the queen would probably move up the board to prevent being taken and threaten the rook and protect the bishop (Qf4 although Qh4 could be an option and still works, even Qf2 which would break my way of solving). If it did unfold like this you have a path to a queen/king fork by (Rxd4+) the king could move out the way but the queen is then lost, so (Qxd4) to take the rook and remove check. you can now pin the queen to the king and threaten taking the queen next move (Rd3), which will lead to an exchange instead of losing queen (Qxd3). then you can play the fork with the knight so the queen is lost (Ne5). from then on it's pretty much a promotion battle. the fork would possibly be the place to stop if a puzzle, rather than try and grind it to checkmate?
Move-by-move commentary?
Understand that combinations cannot be commented on a move-by-move basis. They are conceived as a complete package and only work as such. Anything you write about the 1st move becomes ridiculous when the last move turns out to fail. From an outside viewpoint you can comment on the structure of the combination and how the components are connected which may help you to recognize the pattern in a game but the result will still be that you recognize the potential for that complete combination not for a particular move in it.
Btw, correct move orders are found by brute force calculation. Your mind commonly recognizes the pattern and potential for a combination from experience but you need to visualize all moves and move orders to verify there is one that actually works.
Would someone who can solve this puzzle with relative ease, please offer your thought process as you evaluate the position and determine the correct sequence of moves? It would be so incredibly appreciated. Thank you!
White to move