For a situation like this it is best to think of the energy of the car!The car has a kinetic energy = 1/2mv2 so if the speed has doubled the new kinetic energy of the car will be 1/2m(2v)2 meaning the kinetic energy has increased by a factor of 4. To top the car work must be done by the brakes. The work done will also have gone up by a factor of 4. As work = f x d and the force is the same, the braking distance will have increased by a factor of 4.