Don't read this! a) Rxe7 Qxe7 Qxg6 Bxe5 b) Nxg6+. if Qxg6 Qxg6 Qxg6 Rxe7, if hxg6 Qh4+ picks up the e7 rook, and if Kg8 Rxg7+ Kxg7 Nxe7+ followed by Nxd5 gives White chances, though it will be difficult to win. Fun
Puzzle: This should win, right?
Très bien! Only one small nit: In (b) if ...Kg8 Rxg7+ Rxg7 -+, so ...Kg8 Rxe7! +-. You still did much better than me. My solution was line (a). Darn desperados! ;)
my first thought was1.Nxg6+ hxg6 2.Qh4+ Kg8 3.Rxe7 look better to me, the other thing i notice is if 1...Qxg6 2. Qf3....
Having avoided looking at other posts i came up with
a) This dosnt win as the bishop + K can defend from the two passed pawns and maybe hold the kingside
b)
I found that Nxg6+!! is a win
hxg6(or the rook drops quicker)
Qh4+
And then exchange of Queens with a
R+5p vs B+3p
And white win
Now to read other posts to see the answer!
Edit: Of cource Nxg6!+ can be met by Qxg6 but then Qxg6 looks like enough

(a) Why doesn't 1.Rxe7 Qxe7 2.Qxg6 hxg6 3.Nxg6 Kh7 4.Nxe7 win?
(b) What would you suggest for White instead?
No boards or engines please--visualize the sequence of moves from the initial position.