If you used enough then it would help dampen it a little bit. By that time you would end up spending more then you would on Dynamat or similar material. I also don't think carpet padding will help dampen very much.
There are cheaper alternatives to Dynamat. Elemental Design's eDead is about the cheapest stuff I know of that works well. There is also some other decent products that work for cheaper then Dynamat.
If you want to go an even cheaper route then Lowe's or similar store have some materials you can use. They have some stuff that is for a house that you could use in a vehicle. You could also even go with a spray foam insulation that would help with dampening a bit as well.
When you add the sound dampening products you are just trying to add mass to the panels. That will help to prevent excessive vibration of the panels. Any material you add will help but the best stuff is going to need to be designed specifically for dampening.
Good Luck!!!
