Just count the minus signs.
Each negative flips the sign once, so the result is positive when there is an even number of negatives and negative when there is an odd number. (−2) × (−3) × (−1) has three minuses — odd — so the answer is negative, before you ever work out that it equals −6. The same trick works for division. Count the minuses, ignore everything else, and you already know which way the answer points.
