What is an analytical exposition?

If you are talking about a type of essay, then you need to
break down the type of essay a little bit here.  Expository writing is a traditional
type of essay in which you present facts and opinions with support from examples or
research.  An analysis essay analyzes something.  The "thing" being analyzed may truly
be anything!  You could write an analysis essay on a piece of literature, an historical
event, a type of car, a political policy, a law, a social issue etc.  If you are writing
an expository essay that analyzes something, then you first need to determine your topic
and then more importantly, define your thesis.  Ask yourself what you are trying to
prove about that topic.  Once you know what you intend to prove, then you need to plan
your research and organize your ideas into logical paragraph
divisions.