How is the transfer function determined for a PID controller

I'm reading a tutorial on PID controller, and the transfer function for a mass-spring damper system is given as follows:

where m = 1 , b = 10, k = 20. If a proportional gain is added to the controller, the new tranfser function becomes:

How is the new transfer function derived?

Answer 1

[Sulaymon Eshkabilov]

T = tf(Kp, [1 10 20+Kp]);