I haven't carried out the calculation but I assume that the speed of the particle Q at the point P hits the ground is 4.2 which is in the opposite direction to gravity i.e. upwards. The string will be taut again when the particle reaches this point again and by conservation of energy this means that it will have the same speed but in the opposite direction. Hence:

final velocity, (downwards so positive)

initial velocity, (upwards so negative).

Does it make sense?

If not let me know.