I'm sorry to hear about your loss.
My name isXXXXX am a licensed attorney, and my goal is to provide you with excellent service today.
Yes, you can definitely do a background and credit check on him before you decide whether you want to allow him to purchase one of your mother's cars or not.
You can and should definitely have a contract with him in place if you don't trust him. And, of course, you do not have any obligation to sell the car to him if you don't trust him.
If you have a written contract, signed by both of you, that would cover you in case he breaches the agreement. However, I would definitely suggest that you receive the entire purchase price up front or not give him the car. But if you decide to accept payments, you should leave your name on the title until he fully pays off the car, so you can repossess it if he fails to make full payments.
(This is how all finance companies are able to repossess cars due to missed payments).
You can require that he have a certain amount of insurance
on the car as well, to ensure that you wouldn't be liable in case of an accident
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