Question 165
Find all polynomials $P(x)$ with real coefficients, which satisfy the equation
$$
P(a-b) + P(b-c) + P(c-a) = 2P(a+b+c)
$$
for all real numbers $a, b, c$ such that $ab + bc + ca = 0$.
$$
P(a-b) + P(b-c) + P(c-a) = 2P(a+b+c)
$$
for all real numbers $a, b, c$ such that $ab + bc + ca = 0$.