Multiplication of Negative Numbers

Have you ever though of the meaning of multiplication?
If we are working with positive numbers, the meaning is very apparent.

1 x 2 = 2
2 x 2 = 2 + 2
3 x 2 = 2 + 2 + 2
4 x 2 = 2 + 2 + 2 + 2

It works the same way when we multiply a positive and negative number.

1 x (-2) = (-2)
2 x (-2) = (-2) + (-2)
3 x (-2) = (-2) + (-2) + (-2)
4 x (-2) = (-2) + (-2) + (-2) + (-2)

So what about (-4) x (-2)?
How do we visualize it?
How can we understand this?

I found this question from a maths forum. And have proposed a way to understand this.

Now let us remember this simple rule.
"If we multiply -1 to any term, we are effectively adding it by twice the opposite sign of its number"

Rule:
-1 x a = a + (-2a) = -a

Example :
-1 x 1 = 1 + (-2) = -1
Where "-2" is twice the opposite of "1"


Example :
-1 x -1 = -1 + 2 = 1
Where "2" is twice the opposite of "-1"


What about -4 x -2?

-4 x -2 is actually a convenient way of writing -1 x 4 x -1 x 2

-4 x -2
= -1 x 4 x -1 x 2
= -1 x -1 x 4 x 2 ~ Rearranging the sequence of the numbers
= -1 x -1 x 8 ~ where 4 x 2 = 2 + 2 + 2 + 2 = 8
= -1 x [8 + (-16)] ~ where -16 is twice the opposite of 8
= -1 x -8
= -8 + 16 ~ where 16 is twice the opposite of -8
= 8