Difference of the square - Tricky question for me...

Hi ! First post here !

I'm having some difficulty with a math problem in my math book and I'm starting to think the author made a mistake.

I'm trying to factorize $\displaystyle (a-b)^2 -4$ using the "difference of the squares" technique. According to my book, it should result in the following factors :

$\displaystyle (a-b-2) (a-b+2)$

Now, if I multiply those two factors together, it gives me this :

$\displaystyle a^2 - ab + 2a - ab + b^2 - 2b - 2a + 2b - 4$

then

$\displaystyle a^2-2ab+b^2-4$

so, I'm tempted to conclude that $\displaystyle (a-b-2) (a-b+2)$ does not equal $\displaystyle (a-b)^2 -4$, and therefore that my book is wrong about it.

Am I right ? Did I make a mistake ?