Rereading the question, I think we have all the information we need.
At the starting position, the girl pushes the sled up the hill with speed 4.0 m/s and it slows to a stop uniformly with acceleration 8.0 m/s², then starts coming back down with acceleration 8.0 m/s², and comes to another stop at the bottom of the hill 48 m away from the starting position. If we take up-the-hill to be the positive direction, then we're saying the final position of the sled was -48 m from the origin. The acceleration throughout will have a negative sign.
After being pushed, the sled has a velocity v at time t of
v = 4.0 m/s + (- 8.0 m/s²) t
It comes to a stop when v = 0 :
0 = 4.0 m/s + (- 8.0 m/s²) t
(8.0 m/s²) t = 4.0 m/s
t = (4.0 m/s) / (8.0 m/s²)
t = 0.50 s
In this time, it undergoes a displacement x of
x = (4.0 m/s) t + 1/2 (- 8.0 m/s²) t²
so that, moving up the hill, it covers a distance of
x = (4.0 m/s) (0.50 s) + 1/2 (- 8.0 m/s²) (0.50 s)²
x = 1.0 m
Using the same origin, when the sled begins to slide back down the hill, its displacement from the origin is given by
x = 1.0 m + 1/2 (- 8.0 m/s²) t²
It slides 48 m down the hill from the origin, i.e. - 48 m from the origin, so that
- 48 m = 1.0 m + 1/2 (- 8.0 m/s²) t²
- 49 m = (- 4.0 m/s²) t²
t² = (49 m) / (4.0 m/s²)
t = 3.5 s
So, the sled travels for a total of 4.0 s up then down the hill.