searching a vehicle Weapons, Guns, Firearms

Yes, the police can search a car that you are driving, but don't own.

A few examples: a rental car, a taxicab you drive, an 18 wheeler you drive, a bus you drive, your mother's car (your example).

If the police seek your consent to search, you have the right to say NO.

If you say NO, they could obtain a warrant to search the car.

 

They can also search it with probable cause. Your recourse is to obtain a lawyer and argue their probable cause was not good enough.

 

As above, if you simply assert your right to refuse the search the police may not be able to legally search the car. Often times they are asking your permission because they do not have the information they need to obtain a warrant.