Why does multiplying two negative integers give a positive answer?

Think of multiplication as repeated addition and a negative sign as 'the opposite of'. (−3) × 4 means 3 taken away 4 times, which gives −12. Now (−3) × (−4) means the opposite of (−3) × 4, so it becomes the opposite of −12, which is +12. Another way: on a number line, multiplying by a negative flips the direction. Flip once and you face the negative side; flip again with a second negative and you turn back to the positive side. So a negative times a negative always lands on positive. The simple rule for exams is: same signs give plus, different signs give minus.