5(3+t)-3(t+1) i don't understand how to do this problem.

Okay, just break it up so you're not overwhelming yourself or anything :)
So here, do it in steps-

Let's do 5(3 + t) first:
Distribute the 5, or in non-mathematical terms, just multiply 5 by 3 and t. This will give you 15 + 5t.

Now onto -3(t + 1):
Again, just multiply -3 by t and 1. This will come out to: -3t - 3.

*Note: the 3 is negative because it's "MINUS 3" in the original equation and remember that any number times a negative, if it isn't negative itself, will turn out to be negative (aka "-" x "-" = "+" | "+" x "+" = "+" | "-" x "+" = "-")*

ANYWAY
Now just put both answer together and you get:
15 + 5t -3t - 3. You can only add and minus the LIKETERMS (which are the things that are the same) - in this case, the liketerms are 15 and -3 AND 5t and -3t.

15 - 3 = 12
5t - 3t = 2t

12 + 2t is your final answer. Hope that helps! :)