Why is (n-1)(n+1)=n^2-1?

(n - 1)(n + 1)

The more direct answer is to look at a rectangle with sides of n + 1 and n - 1, let’s go with n = 2

n^2 - 1

xxx

So, that’s a rectangle with sides 2 and 4.

xxxx

If you cut off the end of that rectangle, you make it smaller, 2 * 3 and a separate nub of size 1 * 2

xxx x

xxx

xxx x

The basic answer is that you cross-multiply the values in each factor:

xxxx

n^2 - n + n - 1

Move the nub up to the top:

And you are left with almost a square. One less than the square of side n.