2.Rc8+ is mate very shortly, so it's a better choice.
Where is the mate prfen
My chess program (Fritz) also fails to see the mate very shortly.
What is the continuation of 2.Rc8+ ?
Your wrong the king would play Kf8
Sorry I'm wrong
1.Qh6+ Kg8 2.Rc8+ Kf7 3.Rc7+ Kg8 (everything else leads to instant mate) 4.Rxg7+ Qxg7 5.ef3, when Houdini claims mate in 10 with best play from Black. Anyway, White is a whole Queen up, so this is of no real sgnificance.
After 3. Rc7+, why can't black play 3. ... Kf8 instead of 3. ... Kg8
Qh8 Rg8 Qf6 Ke8 Qe7. As quick and simple as that. But nearly everything wins.
2. Qe6 should be another option for example.