Polarity Question for Powered Home Subwoofer
Is there any way a rookie can figure out what the polarity is?

Yes, however NO GUARANTEES.
In nearly all devices (there can be exceptions) the ground is shared between the power jack and the input. I see what looks like a Cinch input on the right of the photo. One like this:

That (here gold coloured) ring is often connected to ground.
Use a multimeter to measure between that ground and both connections of the DC power input. The connection to the ground of the power input jack should measure a low value close to 0 Ohms. That ground connection should connect to the - of the supply power.
Again: there are no guarantees. The design can be different but usually the procedure above will work.
When measuring, use the Ohms range and measure until the value is stable. Often there are large capacitors connected to the power input. When these capacitors charge, the ohms measurement changes over time while the capacitors charge. If this happens, you're measuring the + connection of the power jack.