Figure[White to move]

Again your inspection of Black’s king shows it to be tightly constrained. Its only flight squares are on the long dark diagonal, which is almost as good as it having no flight squares at all; either way an attack along that diagonal will be a potentially lethal threat. White’s queen has the ability to make such a threat, and it's essential to see that there are two ways to achieve it: Qc3 and Qe5. Neither move works because both squares are defended; but examine the quality of the protection carefully. Qc3+ is made ineffective by Black’s rook on c8. We know a way to address this: White puts his rook on d8 with check. Now what is Black to do?

(a) If he plays RxR, White has succeeded in distracting away the guard of c3. He plays Qc3+ and mates (Black has two futile interpositions which White’s queen eats up).

(b) If Black plays QxR—an important possibility to notice—White still wins by using his queen’s other route to the long diagonal: Qe5+, and again he mates after Black exhausts an interposition (Qf6).

(c) Black’s king has a flight square and is well-advised to use it: in reply to 1. Rd8+ he plays Kg7 and escapes mate. But then White simply plays RxR.

Sometimes, as here, threats meant to overwork a rook on the back rank can end up winning the rook itself, and of course there is nothing wrong with that. You still need to see all the mate threats in play, though, because they are the reason Black is forced to forfeit the rook.