Solve the following quadratic equations by factorization:

We have been given

$x^{2}-x-a(a+1)=0$

$x^{2}+a x-(a+1) x-a(a+1)=0$

$x(x+a)-(a+1)(x+a)=0$

$(x-(a+1))(x+a)=0$

Therefore,

$x-(a+1)=0$

$x=(a+1)$

or,

$x+a=0$

$x=-a$

Hence, $x=a+1$ or $x=-a$.