Determine the acceleration from the first interval.

The object starts from rest, so its initial velocity $u = 0$.
It moves with a constant acceleration, let's call it $a$.
In the first 2 seconds ($t_1 = 2$ s), it covers a distance $s_1 = s$.
We use the kinematic equation: $s = ut + \frac{1}{2}at^2$.

For the first 2 seconds:

$$s_1 = u t_1 + \frac{1}{2} a t_1^2$$ $$s = (0)(2) + \frac{1}{2} a (2)^2$$ $$s = 0 + \frac{1}{2} a (4)$$ $$s = 2a$$

From this, we can express the acceleration $a$ in terms of $s$:

$$a = \frac{s}{2}$$

The "next 2 seconds" means the interval from $t=2$ s to $t=4$ s. So, we need to find the total distance traveled from $t=0$ to $t=4$ s.
Let $S_{total}$ be the total distance covered in $t_{total} = 4$ s.
Using the same kinematic equation with $u=0$ and $a = \frac{s}{2}$:

$$S_{total} = u t_{total} + \frac{1}{2} a t_{total}^2$$ $$S_{total} = (0)(4) + \frac{1}{2} \left(\frac{s}{2}\right) (4)^2$$ $$S_{total} = 0 + \frac{1}{2} \left(\frac{s}{2}\right) (16)$$ $$S_{total} = \frac{16s}{4}$$ $$S_{total} = 4s$$

The distance traveled in the next 2 seconds ($s_{next}$) is the total distance traveled in 4 seconds minus the distance traveled in the first 2 seconds.

$$s_{next} = S_{total} - s_1$$ $$s_{next} = 4s - s$$ $$s_{next} = 3s$$

The object will travel $3s$ meters in the next 2 seconds.