Solution:

Put everything over $8:$ $P(m) = \dfrac{m^8 + m^4 + 2m^2 + 4m}{8}.$ If $m$ is even, every term up top is divisible by $8.$ If $m$ is odd, then $m^2 \equiv m^4 \equiv m^8 \equiv 1$ and $4m \equiv 4 \pmod 8,$ so the numerator is $1 + 1 + 2 + 4 = 8 \equiv 0 \pmod 8.$ Either way $P(m)$ is an integer, so all $4$ values are integers. Therefore, the answer is E.