Finding all of the solutions of the functional equation $P(x+2)+P(x-2) = 8x+6$ 
Let $P(x+2)+P(x-2) = 8x+6$. Find all of the solutions for $P(x)$.

If we put $P(x) = ax + b $ the answer is obvious but how to determine all of the solutions?

This difference equation is linear so it's solution can be reprasented as
$$
P(x) = P_h(x) + P_p(x)
$$
with
$$
P_h(x+2)+P_h(x-2) = 0\\
P_p(x+2)+P_p(x-2) = 8x+6
$$
for the homogeneous we propose $P_h(x) = \gamma^x$ and after substituting
$$
\gamma^x\left(\gamma^2+\frac{1}{\gamma^2}\right)=0
$$
and the solutions are
$$
\gamma_h = \left\{-\frac{1+i}{\sqrt{2}},\frac{1+i}{\sqrt{2}},\frac{1-i}{\sqrt{2}},-\frac{1-i}{\sqrt{2}}\right\}
$$
so the homogeneous solution is 
$$
P_h(x) = C_1\sin\left(\frac{\pi x}{4}\right)+C_2\cos\left(\frac{\pi x}{4}\right)+C_3\sin\left(\frac{3\pi x}{4}\right)+C_4\cos\left(\frac{3\pi x}{4}\right)
$$
now for the particular making $P_p(x) = 4x+3$ we conclude
$$
P(x) = 4x+3+C_1\sin\left(\frac{\pi x}{4}\right)+C_2\cos\left(\frac{\pi x}{4}\right)+C_3\sin\left(\frac{3\pi x}{4}\right)+C_4\cos\left(\frac{3\pi x}{4}\right)
$$
