Sean asks for an explanation of why a negative times a negative is a positive.
Here we go...
-x * -y = (-1 * x) * (-1 * y) = (-1 * -1) * (x * y) = 1 * (x * y) = x * y
Obsessive Update: The above assumes -1 * -1 = 1. Why? The only thing I thought of, which really becomes a general rule is to create a contradiction. If -1 * -1 = -1 then...
...which is the contradiction we're hoping for.
-1 * (1 - 1) = (-1 * 1) + (-1 * -1)
-1 * (0) = (-1) + (-1)
0 = -2
Update: maybe Sean is looking for an applied explanation rather than a proof?
Hmm... that would be a good one. When do we do -x * -y in the real world?
Well, I do x * -y in the real world when I pay my mortgage (-y) for some number of months (x). So maybe I could ask how much I would have had had I not paid my mortgage (-y) the previous number of months (-x).
Clearly the result of these two formulas should be the opposite. If I paid -$N USD then if I didn't I would have saved $N USD.