Find all polynomials $P(x)$ with real coefficients such that for all real numbers $a, b$, and $c$:

$$
P(a+b-2c) + P(b+c-2a) + P(c+a-2b) = 3P(a-b) + 3P(b-c) + 3P(c-a)
$$
For $a=b=c=0$, we get $P(0)=0$. For $a=b$, it follows that $P(2 b-2 c)+P(c-b)+P(c-b)=3 P(0)+3 P(b-c)+3 P(c-b)$. By setting $x=b-c$, we obtain $P(2 x)=3 P(x)+P(-x)$. Let $a_{n}$ be the leading coefficient of $P$. The previous equation leads to $2^{n} a_{n}=3 a_{n}+(-1)^{n} a_{n}$, which implies that $n=1$ or $n=2$. We then seek solutions of the form $P(x)=a x^{2}+b x$ and find that only $P(x)=x$ is suitable.