When you use F=ma ... you need the NET force acting on the object.
There are two parts to the object's motion here:
(I) In the first four seconds, the object has some net force, hence it has some acceleration (given from F=ma) based on that net force (and the mass, of course). What is the velocity of the object at the end of this first phase of motion?
(II) There is a third force now acting on the object. The initial velocity for this phase of motion is the same as the velocity you found in part (I). You are told that the object displaces 2m in the same direction as its motion before (so, 2m RIGHTWARD). You also know that the final speed at the end of this phase is 1.24 m/s but this implies that the final velocity is either +1.24 or -1.24 ... both possibilities with lead to two possible forces F. In either case, you should introduce F acting rightward and determine what it is ... if it turns out to be negative, it means that it acts to the left instead. In order to find F, you need to use F=ma again .... what is the acceleration of the object in phase (II) ? Use the displacement, initial/final velocities in SUVAT to work that out.
EDIT: Based on the question's wording, I think they only want you to use +1.24 for the final velocity for (II). There is ambiguity in their phrasing when they say "moved a further 2m in the same direction" ... does that mean total distance travelled is 2m rightward, or does it mean the displacement from the end of phase (I) is 2m rightward?
Anyway ... choose +1.24 for first attempt and see what force you get.