Why is my battery life so short

I recently was given a 2010 13” macbook pro, and upgraded it to High Sierra 10.13.6. I didn’t have it on the old operating system for long enough to really notice the battery life, but currently I’m only getting about 2 or 2.5 hours. I know that this machine used to get about 6-7 when new.

I checked the power section of system info and this is what I have:

I’ve also noticed that the system tends to warm up quite a bit, but I never hear the fan going, so maybe thats an issue?

I am comfortable opening this up and operating on it, as well as I am willing to purchase a new battery for it, bu I would like to know with a fair amount of certainty that purchasing a new battery will solve the issue before I spend the money.

Any help would be greatly appreciated! Thanks!