x = 6 is correct
Focus on the right triangle EDF. The short leg is x = 6. The hypotenuse is 4+8 = 12. Since the hypotenuse is twice that of the short leg, we have a 30-60-90 triangle. Recall that the ratio of the short leg, long leg and hypotenuse is 1:sqrt(3):2 or you can think of it as x: x*sqrt(3): 2x. The 30 degree angle is opposite the short leg (smallest angle opposite smallest side). So the 60 degree angle is angle FED, which is the angle formed by the two chords.
This means angle BEC is 120 degrees since 120+60 = 180.
Now use the second part of the hint which says that the two arcs add up and are cut in half to get the angle formed by the intersecting chords. This means,
angle BEC = (1/2)*(arc BC + arc AD)
120 = (1/2)*(arc BC + arc AD)
2*120 = 2*(1/2)*(arc BC + arc AD)
240 = arc BC + arc AD
arc BC + arc AD = 240 degrees
