Given:
Initial velocity (u) = 0
Acceleration (a) = 4 m/s²
Time taken for acceleration (t1) = 4 s
Time taken for retardation (t2) = 8 s
Final velocity (v) = 0

(a) Average speed:
To find the average speed of the particle during the entire motion, we need to first find the total distance covered by the particle. We can divide the motion of the particle into two parts - the first part where it is accelerating and the second part where it is retarding.

For the first part:
Using the equation of motion v = u + at, we can find the final velocity of the particle after acceleration:
v1 = u + a*t1 = 0 + 4*4 = 16 m/s

Using the equation of motion s = ut + 1/2 at², we can find the distance covered by the particle during acceleration:
s1 = ut1 + 1/2 a*t1² = 0 + 1/2 * 4 * 4² = 32 m

For the second part:
During the second part, the particle is retarding with a constant acceleration. We can find the magnitude of the acceleration using the equation of motion v² = u² + 2as, where v = 0, u = 16 m/s and s is the distance covered by the particle during retardation. Solving for s, we get:
s2 = u²/2a = 128 m

Thus, the total distance covered by the particle is:
s = s1 + s2 = 32 + 128 = 160 m

The average speed of the particle is given by the formula:
Average speed = Total distance covered/Total time taken

Total time taken = t1 + t2 = 4 + 8 = 12 s

Therefore, the average speed of the particle is:
Average speed = 160/12 = 13.33 m/s

(b) Average velocity:
To find the average velocity of the particle during the entire motion, we need to consider the direction of motion. Since the particle starts from rest and comes to rest, the displacement of the particle is zero. Therefore, the average velocity of the particle is also zero.

Explanation:
- Divided the motion into two parts.
- Used equations of motion to find the distance covered during acceleration and retardation.
- Used the formula for average speed to find the average speed of the particle.
- Explained that the displacement of the particle is zero, hence the average velocity is also zero.