You might get to advance it a rank or two, but you're THREE away from queening, and I don't see you getting that far.
20. hxe4 Qxg4+
21. Qg2 Qxg2+
22. Kxg2 c5
23. Bb5 c4
24. Nd2 Ne6
25. Nf3 c3
26. Ra2 Rac8
27. bxc3 Rxc3
28. Bd2 Rb3
29. Bxb4 axb4
30. Kf2 Rc3
31. Rb1