44 Rxh7+ Rxh7
45 Qd4+ Rg7
46 Rh4#

The second-best move order would probably be

44 Qd4 Qd8
45 R(f4)h4 Qxh4
46 Rxh4

44.Qd4 pinning the Rook on g7
<if...g5 45.Rxf5 winning a free Bishop

Black threatens Qxc5.

White to play (44.?) "Easy"

Quite often the easiest way to get at the enemy king is to sac a piece to open up the position.

44. Rxh7+! Rxh7

There is a quicker mate after 44...Kxh7 45. Rh4#.

<44.Rxh7+>

(1) 44...Rxh7 45.Qd4+ Rg7 46.Rh4#

(2) 44...Kxh7 45.Rh4#

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Good point. I thought Black loses straightforwardly after ... Kf8 as well, since White can continue with Rxf5+/Rf6+, but I haven't quickly seen how to beat the defense:

43 ... Kf8
44 Rxf5+ gf
45 Rf6+ Ke8
46 Qh6 Qe1

Agreed. The text line is forcing whereas my line takes longer if Black becomes mated or ends up in big material advantage for White as mentioned by <Cheapo by the Dozen>. I missed 44...Qd8 and played 44...Qxc5. The latter ends up in mate and the former(...Qd8)ends up loosing the Queen