In these kind of problems, I recommend redrawing the triangle after deriving some preliminary information.

Two properties we will use here:

1) Angles on a line are supplementary (their sum is 180)
2) Sum of angles of the triangle is 180

Since Angle ADC is 120 => angle ADB = 60 (sum must be 180). Since Angle BAD is 30, the angle ABD (or angle ABC) = 90.
Now that we know that angle ABC is 90 - lets redraw the triangle to make it look better.
To get the area of triangle ADC, we just need to figure out "h" and the area would be 1/2 * h * 4

Finding h is very easy for the 30, 60, 90 , h = 6* sqrt{3}

Area = 1/2 * 4 * 6 * sqrt{3}

Area = 12 * sqrt{3}

(D)