Watch your formulas
You wrote (ab)^2 = a^2 + 2ab  b^2 but that is wrong. Was it like that all the time and I did not notice before?
(ab)^2 = a^2  2ab + b^2
On the other hand, I agree that (a + b)^2 = a^2 + 2ab + b^2
I see that as the square of a sum being equal to the sum of two squares plus two rectangles. If I have a square garden with sides measuring a (area = a^2), and I want to make it bigger, I can add two rectangles of length a and width b to the South and East sides (area = ab for each one) and a square of side b (area = b^2) to fill the Southeast corner. I end up with a square garden of side a+b. The area is (a+b)^2, and equals the sums of the areas of the two squares and two rectangles listed above.
