What is the power of the engine when the velocity of the car is v, mass m, acceleration a, and external resistance R

The correct option is (R+ma)v.

To find the power of the engine, we must determine the driving force that the engine needs to exert to maintain the car's acceleration while overcoming resistance.

According to Newton's second law of motion, the net force acting on the car of mass $m$ to produce an acceleration $a$ is:
$F_{\mathrm{net}}=ma$

The actual force exerted by the engine ($F_{\mathrm{engine}}$) is opposed by the external resistance ($R$). Therefore, the net force is the difference between these two forces:
$F_{\mathrm{net}}=F_{\mathrm{engine}}-R$

By equating the two expressions for the net force, we can solve for the engine's force:
$F_{\mathrm{engine}}-R=ma$
$F_{\mathrm{engine}}=R+ma$

Power ($P$) is defined as the rate at which work is done, which is the product of the driving force and the velocity ($v$) of the car:
$P=F_{\mathrm{engine}}·v$
Substituting the expression for the engine's force into the power formula gives:
$P=(R+ma)v$