Do not understand formula... How does;
$x^{n} -y^{n}=(x-y)(x^{n-1} + x^{n-2}y+...+x y^{n-2}+ y^{n-1} )$
work on $x^{2} - y^{2}$ 
When I attempt to apply the formula on $x^{2} - y^{2}$ 
I get the following
$x^{2} - y^{2} =(x-y)( x^{1} + x^{0}y+...+x y^{0} + y^{1} )$
$x^{2} - y^{2} =(x-y)(2x+2y)$
which is obviously false. What is the correct way to use the formula?

If you use proper notation, instead of dots, you get the correct result:
$x^n-y^n=(x-y)\sum\limits_{k=0}^{n-1}x^{n-1-k}y^k$
Then $x^2-y^2=(x-y)(x^{1-0}y^0+x^{1-1}y^1)=(x-y)(x+y).$
