Solution

The regular expression is

$a^*b(a+b)$

It means:

Any number of $a$'s, followed by one $b$, followed by either $a$ or $b$.

So, strings accepted are like:

$ba,\ bb,\ aba,\ abb,\ aaba,\ aabb$

Minimum DFA states needed:

State $q_0$: Initial state, reads any number of $a$'s.
State $q_1$: After reading required $b$.
State $q_2$: Final state after reading last symbol $a$ or $b$.
State $q_d$: Dead state for extra or invalid symbols.

Thus, total minimum states are

$4$