Factorize:

We have:

$a^{2}+a b(b+1)+b^{3}=a^{2}+a b^{2}+a b+b^{3}$

$=\left(a^{2}+a b^{2}\right)+\left(a b+b^{3}\right)$

$=a\left(a+b^{2}\right)+b\left(a+b^{2}\right)$

$=\left(a+b^{2}\right)(a+b)$