What led Portugal, Spain, France, and England to invest in expeditions and what territory did they claim?

To some
degree, all of these countries had the
same motivations for investing in expeditions that went
out to explore the
world.  They might have had different mixes of motivations, but the general

set of motivations was the same.  Historians tend to use the shorthand expression God,
gold,
and glory as a way of saying what those motivations were.

The God part of
this expression means that Europeans felt that they
needed to explore in order to spread
Christianity.  In those days, Christians
tended to believe that anyone who was not Christian
would be condemned to
damnation in the afterlife.  Therefore, they believed that God wanted them
to
bring the Gospel to people who had not heard it.  The Portuguese and Spanish probably
felt
this motivation more strongly than the French and, particularly, the
English, but it was present
for all nations.

The gold part
of the expression refers to getting rich. 
The Portuguese started exploring
Africa in hopes of gaining a sea route to the spice islands of
Asia.  They
hoped to get wealthy in that way.  After the Spanish discovered the Americas
and
the silver and gold mines there, they got a great deal of wealth.  Other
countries wanted part
of that.  They hoped to find new lands that had
resources too.

The glory
part of the expression can refer
either to personal glory or to the glory that came to countries
with large
empires.  Europeans felt that having a large empire meant that their country
was
great.  Therefore, they wanted to explore and find places that could
become part of their
empires.  They did not want to lose out in this
competition, thus making themselves look weak
and unimportant. 

As for what territories were claimed, this varied over
time.  The
best thing to do is to look this up in your textbook to see what it says. 
Otherwise,
you can Google each countrys empire.  One problem with that is
that different countries claimed
different territories at different times. 
Another problem is that the list for each country can
be very long.  For
example, do you need to know that France claimed St. Pierre and Miquelon and

that England claimed the South Sandwich Islands?  My guess is that your text has a list
of
important territories claimed at a given time.  I would suggest that you
find that list and use
it.