Check is like revise something. Check out is when you leave a hotel, for example, and you give the keys back

They both have a wide variety of meanings and both nouns and verbs. A check can be a piece of paper used to transfer bank funds, your bill at a restaurant, or a mark that looks like this: √. "Check out" can be the place in a grocery store where you pay for your purchases. "To check" can be to verify, examine or investigate something; or type or write a check mark. It can also mean to bring something to a stop. "To check out" can be to leave a hotel after your stay, or to pay for your purchases in a store. It can also be to investigate something, to explore a place, or to look at someone or something.

^^^ meanings as nouns and verbs ^^^

Check is to to see if something is ok, for example, I have to check and see if I turned off the stove. You can also check something that you have written for errors. Another way it could be used if someone asks you if you have a certain item and you can say, I will check and see if I still have it. Check out is usually used to check out of a hotel, meaning that you pay your bill and leave. In slang "check out". can mean you stop paying attention to something or you are not interested in it anymore. Example: When his wife starts nagging him he just checks out. (stops listening and thinks about something else other than her and her nagging. Also I guy can "check out an attractive woman or vice versa. Look them up and down. Ex. Check out that good looking chick over there.

to check - to verify to check out - to leave (hotel/school) or to observe (something)