Factorize each of the following expression:

$a^{2}-b^{2}+a-b=\left(a^{2}-b^{2}\right)+(a-b) \quad$ [Grouping the terms]

$=(a+b)(a-b)+(a-b)$

$=(a-b)(a+b+1) \quad[$ Taking out the common factor $(a-b)]$