Final answer:
A Self-Service Car Wash would not be held liable for damages to a vehicle in the absence of a bailee agreement, as the owner retains control of the vehicle while using the service.
Step-by-step explanation:
The establishment that would NOT be held liable for damages to a vehicle while in the business' care due to the lack of a bailee agreement is Option 4: Self-Service Car Wash. A bailee agreement is a contract where the bailee (the party temporarily holding property) promises to return the property or to otherwise handle it in a certain way.
In the cases of an auto repair shop, valet parking service, and car rental agency, there is an implied bailee relationship where the businesses take possession and control of the vehicle, thereby owing a duty of care. However, at a self-service car wash, the business does not take possession of the vehicle; the owner maintains possession and control of the vehicle at all times.