The rule of how to multiply negatives is simple: 

To multiply a negative number by any number, take the (positive) opposite of the negative number and multiply it by the other number, and take the opposite of that

Negative by positive number

For a case where a negative number is multiplied by a positive, the rule above can be restated as: take the positive number, and the negative WITHOUT the minus sign, multiply them and add the inus sign to the product. 

Example: multiply 5 and -3.  Take 5 and -3 without the "-" sign: 5 and 3  Multiply them: 5*3 = 15.  Add the minus sign: -15.  Conclusion: Product of a negative and a positive is always a negative number

Negative by negative number

When you have to multiply two negative numbers, such as (-5) * (-3), the rule above should be applied twice:  (-5)*(-3) is the opposite of 5*(-3), which is the opposite of the opposite of 5*3.  5*3 is 15  Opposite of 15 s -15.  Opposite of opposite of 15 is opposite of -15, and opposite of -15 is 15 again.  Conclusion: Product of a negative multiplied by negative is always POSITIVE

Here's a handy table: sign of result when we multiply

Multiply: Positive Negative
Positive + -
Negative - +

Examples:  2*3=6 2*(-3) = -6 (-2)*3 = -6 (-2)*(-3) = 6 

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