Particle 1 has a velocity v1 = 2 m/s and a mass m1 = 2 kg . this particle collides with particle 2 of mass m2 = 6 kg , which is at rest, and then the two particles stick together. the speed of the conglomerate particle is

This follows the law of conservation of momentum. Momentum is the product of mass and velocity of object. 
Momentum = mass(m) x velocity(v) 
law of conservation of momentum means that the total momentum of system before the collision of 2 objects is equal to the total momentum after the collision 
Before the collision total momentum 
= m1v1 + m2v2
m1 = 2 kg
v1 = 2 m/s
m2 = 6 kg
v2 = 0 m/s
substituting the values in the equation 
total momentum before = (2 kg x 2 m/s) + (6 kg x 0 m/s)
total momentum = 4 kgm/s
after the collision the 2 objects stick together and have a common velocity 
total momentum after the collision  = (6 kg + 2 kg)x V  = 8V
V = speed of the conglomerate particle

since total momentum before is equal to total momentum after
8V = 4
V = 2 m/s
speed of conglomerate particle is 2 m/s