When multiplying more than two integers how can you determine the sign of the product

When you multiply two numbers, let's say x and y, there are no negatives, so you know that it is positive. So that means it's xy. Now, imagine we make that a - x times y. Since there is only 1 negative, it is negative. So that would be - xy. Now, imagine you have - x times - y. Since there are an even number of negatives, they cancel out! (Magic). So - x times - y is xy.