**Position of the particle**
The position of a particle moving along the x-axis is given by the vector r = |2t^2 - 40t| * icap m, where t represents time and icap represents the unit vector along the x-axis.

**Finding the speed of the particle**
To find the speed of the particle, we need to calculate the magnitude of the velocity vector. The velocity vector is the derivative of the position vector with respect to time.

v = dr/dt
= d(|2t^2 - 40t| * icap)/dt
= (d(2t^2 - 40t)/dt) * icap

Since the derivative of |2t^2 - 40t| with respect to t can be positive or negative, we take the absolute value to find the magnitude of the velocity vector.

|v| = |d(2t^2 - 40t)/dt|
= |4t - 40|

**Finding the time at which speed is 4 m/s**
To find the time at which the speed of the particle is 4 m/s, we equate the magnitude of the velocity vector to 4.

|v| = 4
|4t - 40| = 4

We can split the equation into two cases, considering the absolute value.

Case 1: 4t - 40 = 4
4t = 44
t = 11

Case 2: -(4t - 40) = 4
-4t + 40 = 4
-4t = -36
t = 9

Therefore, the particle will have a speed of 4 m/s at t = 11 and t = 9.

**Explanation**
The given position vector represents the position of a particle along the x-axis. The magnitude of the position vector is given by |2t^2 - 40t|. To find the speed of the particle, we need to calculate the magnitude of the velocity vector. The velocity vector is the derivative of the position vector with respect to time. By taking the derivative and finding its absolute value, we obtain |4t - 40|. To find the time at which the speed is 4 m/s, we equate the magnitude of the velocity vector to 4 and solve for t. The resulting values of t are 11 and 9. Therefore, the particle will have a speed of 4 m/s at t = 11 and t = 9.