Which one of the following regular expressions correctly represents the language of the finite automaton given below?

Let the start state be $q_0$, the top final state be $q_1$, and the bottom final state be $q_2$.

1. From $q_0$, transitioning to $q_1$ requires symbol $a$. While at $q_1$, reading any number of $b$'s keeps the automaton in $q_1$ (represented by $b^*$). A subsequent $b$ takes it back to $q_0$. Thus, a round trip from $q_0$ to $q_1$ and back to $q_0$ is given by $a b^* b$.

2. From $q_0$, transitioning to $q_2$ requires symbol $b$. While at $q_2$, reading any number of $a$'s keeps the automaton in $q_2$ (represented by $a^*$). A subsequent $a$ takes it back to $q_0$. Thus, a round trip from $q_0$ to $q_2$ and back to $q_0$ is given by $b a^* a$.

3. Any string that returns to $q_0$ zero or more times is generated by the regular expression:

   $$(ba^*a + ab^*b)^*$$

4. To be accepted, the path must finally end in either:

   • State $q_1$, which contributes an additional $a b^*$

   • State $q_2$, which contributes an additional $b a^*$

Combining these, the complete regular expression is:

This matches option D.