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 country’s 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.