j’ai, tu as, il a … fini

nous avons, vous avez, ils ont … fini

In your example, nous avons, vous avez, ils ont … espéré.

Never with the avoir verbs would the past participle agree with the subject. We would never write “elle a esperée.”

However, when the past participle is being used in the passive sense, as a modifier of a noun, it will agree with the noun it modifies. For example:

La naissance d’une fille est tellement esperée par la famille. (Note that esperée is agreeing with naissance; if the family were hoping for a boy the participle would be the same, as in “la naissance d’un fils est tellement esperée par la famille.”

Little note to mademoiselle Kim: it’s easy to install a second language pack. There are several French ones. You’d get all the accents. The right-hand keys on the keyboard would change, you could either make a little map or memorize them or peck around. The change is not permanent, it’s only while you are in the 2nd language mode. After that you flip back to English USA & keyboard goes normal again.

If you have windows or vista, click in control panel on “language & regional options.” This will take you on to choosing a 2nd language pack.

Speaking of French accents, our student here is doing famously well with them ! Mention très bien !