Since it's a transverse wave, a particle on the string moves left and right as the wave passes by, but the particle doesn't travel forward or backward at all.
So the little red dot moves 'A' to the left, then 'A' back to the center, then 'A' to the right, then 'A' back to the center again.
All together, the red dot moves a total distance of 4A . (choice 'a')