How to play this?

Ke5 would have been a good start.

Keep the Black King away from his own pawns and try to pick them off.

I think black wins regardless, but I agree that Ke5 is white's best shot.

Ke5 h5xg4 forces

f3xg4 and I think Kc6 is black's move as fxg4 is useless as white's h stops both black's gs, and the more obvious Kd7 draws, I think.

Now white is in a quandary.  His king cannot take the backward of two connected passed pawns or the forward pawn queens.  He must not permit black king to d6 or black's e will advance and policing it will give black's king more range.White's best try is something like h3 and hope for black's g5

h3 g5?

gxf and the rest is obvious.  However,

h3 Kc5 now i think his best shot is g5 creating a real threat of the h pawn advancing forcing black's g to capture and white's g queening.

g5 f4 is an interesting pawn sac which I think wins.

Please take someone else's advice over mine,  I tend to be imprecise.

In fact Ke5 doesn't draw easely

First play gxf5 which forces a gxf5 recapture, then play Ke5 for the draw

wouldn't gh force ...gh, splitting the pawns and allowing white to get Ke5 as ... Kd6 would allow a queen.  Anyway, it just seems preferable to running Ke5 first to net a loss of pawns 2 for one or advancing the g pawn, which allows the black king to take opposition, which stops the white attack dead in its tracks.

I think gh gives the maximum amount of play, but I have no idea who wins or whether white should really be praying for a draw.

gh gh

Ke5 (...)

Nothing can prevent  Kf6 and then Kg5. 

Black king must protect e pawn while you can menace h pawn.

Close.

Pawnfork and Sothilde,

Both your initial pawn exchanges lead to the same positions. So,

I tried these positions on Chessmaster10 expecting some clarity as it is usually good with endgames. From starting position, ChessMaster played white and agreed with you. I was black, and admittedly mediocre with endgames.

1 gxh5 gxh5

2 Ke5 Kd7

3 Kf4 Kd6

4 h4 Kd5

5 Kg5 Ke5

6 Kxh5 Kf4

7 Kg6 e5

8 h5 e4

9 fxe4 fxe4

10 h6 e3

11 h7 e2

12 h queens  e queens

13 Qg7 Qe4+

14 Kf7 and I assumed draw

I had previously played ChessMaster with myself as white and it seemed to confirm my previous analysis:

1 Ke5 hxg4

2 fxg4 Kd7

3 h3 Kc6

4 g5 Kd7

5 h4 Ke7  an immediate h5 would be meaningless so I tried:

6 Kf4 Kd6

7 h5 e5+

8 Kf3 gxh5

9 Kg3 e4

10 g6 Ke7

And I really don't think white could play much better.  Pawnfork's and Sothilde's points hold, unless someone can show otherwise.