a2 – b2 = (a+b) (a-b) is indeed true.
First let's look at this numerically. At least for an initial level of verification. Let's take for example a = 9, and b = 4.
92– 42 = 81 – 16 = 65 or putting it into the terms we'd like to use: a2 = 81 (with a=9) and b2 = 16 (with b=4) and finally (9 + 4)(9 – 4) = 13 * 5 = 65! WOW! It really works! Try it out with several other numerical examples.