Any electronics experts out there?

I’m trying to revive a cheap Lepai LP-2020A+ amplifier that I’m sure I picked up at the Goodwill Outlet. It won’t power on, but if I jump a couple of connections on the relay it will start up. A bit of research has shown that capacitor C15 may be installed backwards, but I can’t find any polarity indications on the board (unless I just don’t know what I’m looking at):

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Illustration for article titled Any electronics experts out there?
Illustration for article titled Any electronics experts out there?
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Illustration for article titled Any electronics experts out there?

The other thing that concerns me is the soldering on the amplifier chip itself:

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Wildly inconsistent, and apparently missing solder on one of the connections. There were some sloppy bits of solder debris on the board that popped right off as well, although they did not appear to be shorting anything out.

Also of concern is the condition of capacitor C1. It’s not popped, and voltage flows across it, but it looks beat to hell. I’m assuming that it was probably too much heat during assembly:

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Illustration for article titled Any electronics experts out there?

I’m used to capacitors like this being mounted flush to the board, but that obviously isn’t the case here, not that that’s really an issue, just poor workmanship. The apparent heat damage is a little bit of a concern, but that’s probably just poor workmanship again.

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I can get the unit to power on if I jump a couple of pins on the relay, but that’s not a permanent fix, just a troubleshooting step. I haven’t checked for audio output when I do that, but figured that it wouldn’t matter since the real problem lies elsewhere.

So, how do I know if C15 is installed backwards? Should I complete the soldering on the amplifier chip? Anything else I should be looking at? Thanks in advance