This one uses the Myhill-Nerode theorem.

Given a language L = {w {a, b} : w starts and ends with the same symbol and has the same quantity of a's and b's} , and the following pair of strings x and y:

{bbbabaaa, babbaab}

{bbbabaaa, bbaabbaba}

{bbbabaaa, abbbabaa}

{babbaab, bbaabbaba}

{babbaab, abbbabaa}

{bbaabbaba, abbbabaa}

define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L for each pair above.