The negative speaker wire broke off the board how can I fix it

grimesjosh1423  not sure what you got there but it looks like the one on the right is your cable. It looks like someone tried to solder the insulation too. The first thing you want to do is to remove the insulation from the cable. Next you want to use some flux and desoldering wick and clean that blob of solder on the left. After that you can solder the cable from the right to the pad on the left with some flux and solder. This seem the be a though-the-hole construction and we do not know what the cable looks like on the other side but it is conceivable that you can feed some more wire to the solder pad by pushing the wire through. If not you can always solder a longer wire on the short end.