Algebra 2 question, how many friends made the trip?

Let the original number of friends be F, then the original cost per person was 300/F

When 3 more friends join the trip, the number going is (F + 3) and the cost per person is 300/(F + 3)

But we are told that, [300/(F + 3)] + 5 = 300/F . . . multiplying throughout by F(F + 3) gives :-

300F + 5F(F + 3) = 300(F + 3)

300F + 5F² + 15F = 300F + 900

5F² + 15F - 900 = 0

F² + 3F - 180 = 0 . . . which factors as (F + 15)(F - 12) = 0

As we are only concerned with the positive number solution then F = 12.

The original cost would have been 300/12 = $25 each and the cost after 3 extra people became involved would be 300/15 = $20.

There's not enough information to answer the question. You need to know how much the cost was after 3 friends joined the trip. Just saying that it's 5 dollars less per person doesn't help because you don't know how large the original group is.