How to remove odd favorites listing on address bar?

On the far right of the address bar (Explorer 8), is a down-pointing arrow
which when clicked, lists recent browsing history, and then (on my computer)
a very incomplete list of my favorite sites.

Is there a way to delete or edit this outdated and incomplete list of
favorites? My computer automatically deletes the history when I sign out of
Windows, but that darn favs list remains.

Thanks for your assistance.