Let's do some surgery and disect this problem!

Two different people can drive, so this gives us two possibilities. Of those two possibilities, there are different ways you can arrange the other four people. This applies to Person A and Person B, so we have ways that the family can be arranged. Lastly, we can have two or three people in the front, so this once again doubles out possibilities to give us a final answer of arrangements.

Let me know if I missed anything!