Good page. Helped me a lot with the geometric representation.
Those 3 basic forms
are also the basis for mental arithmetic. Lets take your example:
If you wanted to multiply 38 * 42 you could use that form with y=10.
Answer 16*100 - 4 = 1596 like magic.
For 63 * 57
63*57 = 36 * 100 -9 = 3591
With a little practice and algebra, you can do lots of arithmetic in your head.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.