Which one of the following regular expressions correctly represents the…

Key idea: decompose any accepted string into a sequence of small building blocks that mirror moves between the start and the two accepting states.

• Final segment: any accepted string must end by moving from the start to one of the accepting states and then taking zero or more loop symbols there. Those two possibilities are ab* (start → top accepting, then any number of b's) and ba* (start → bottom accepting, then any number of a's). So the final part is (ab* + ba*).

• Round trips: each time the automaton leaves and then returns to the same phase it consumes a block of the form ba*a (go from bottom-side into top-side and back) or ab*b (go from top-side into bottom-side and back). These two block types are combined as (ba*a + ab*b).

• Putting it together: you may repeat those round trips zero or more times and then finish with the final segment. That yields the regular expression (ba*a + ab*b)*(ab* + ba*).

Therefore the correct regular expression for the language of the given automaton is (ba*a + ab*b)*(ab* + ba*).