How many solar panels would one need to carry to fully charge Tesla model S or S in 20 minutes?
Using the sun as energy source is actually one of the few reasonable ways to overcome the issues we are globally facing. The problem is the efficiency of conversion from sunlight to electricity (the conversion efficiency of photovoltaic modules is about 20%), plus the energy storage (Expensive, heavy batteries with limited lifetime obviously can't be the last word). 

Anyway:
According to Wikipedia, the Tesla S battery has a capacity of at least 60kWh.
Charging 80% (48kWh) within 20 minutes requires a power of 48kWh * (60/20h) = 144kW.

One of the rare 400W panels I've found is from LG. It has a length of about 2 m and a width of about 1 m, so about 2 m². It weighs about 21 kg. 
Assuming that the sun is strong enough for the panels to work at their peak power, this would result in 360 panels, or 720 m². The weight of just the panels is 7.5 tons.

This is just a rough estimation, which doesn't consider many things (efficiency of the charger, weight of the mechanical stuff, etc.), but I believe


  Would carrying a folding device that stacks solar panels in a 3 meter
  by 2 meter 2 wheel trailer be efficient


can be answered quite clearly with No.