Solution:

If there are 6 tangent lines, then there are 12 unbounded regions.

It is not very difficult to visualize that each tangent line drawn creates two unbounded regions.

The animation shows 3 tangent lines. The unbounded regions are in yellow and the bounded regions in green.
For 3 tangent lines, there are 6 unbounded regions.

Food for thought:

How do you arrive at the solution to such a problem?
By induction?!
Try to solve the problem for 1 tangent line, 2 tangent lines, and so on. It is easy to then generalize.