(a+b)(a+b)  &  (a-b)(a-b)


A square of side lengths a + b
It's area is (a+b)(a+b)
This area has 4 parts:

The lower left b x b square.
The upper left a x b rectangle.
The lower right a x b rectangle.
The upper right a x a square .

So (a+b)^2 = a^ + 2ab + a^2

 
 
A square of sides b-a.
It's area is (b-a)^2.
The animation below shows how this is the same as the illustration to the right.
b^2 - 2ab + a^2.
To show the rectangles are negative,
they're colored black

 
The negative black area can be thought of anti matter. As it meets the white area, they annililate each other. 

After it's over,
only a square of sides (b-a) is left.

So (b-a)^2 = a^2 -2ab + b^2
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