Answer
See answer.
Work Step by Step
Consider a volume of fluid in the pipe.
At the beginning and at the end of the pipe, there is a force toward the enclosed fluid, given by the pressure multiplied by the area, $F=PA$. Let the pressure at point 1, the beginning, be larger.
The area of the pipe is constant, so the net force is $F_{net}=(P_1-P_2)A$.
The power required is the force multiplied by velocity v.
$$Power\;needed=F_{net}v=(P_1-P_2)Av $$
However, the cross-sectional area A multiplied by the velocity v is Q, the volume rate of flow.
$$Power\;needed= (P_1-P_2)Q $$
