so there are two forces acting on the P, a resorting force and an additional force that is forcing the system to oscillate.

There is only one force acting on the particle: the tension (or compression) in the spring.
the resorting force is due to the tension in the spring.

by hookes law.

we are giving that the displace

due to the addition force is

so the acceleration due to this force at time

is

(by taking the derivative with respect to time twice)

Sorry, but that is nonsense. You can't just differentiate what's happening at one end of the spring in prder to find out what's happening at the other end.