Let $P(x)$ be a nonzero polynomial with real coefficients such that

$$
P(x)=P(0)+P(1) x+P(2) x^{2}
$$

holds for all $x$. What are the roots of $P(x)$ ?
Let $c=P(0)$. Selecting $x=1$ in the given, we have that $P(0)+P(1)+$ $P(2)=P(1)$, so $P(2)=-c$. Selecting $x=2$, we find that

$$
-c=c+2 P(1)+4(-c) \Longrightarrow P(1)=c .
$$

Therefore, $P(x)=c+c x-c x^{2}$. As $c \neq 0$, we solve $1+x-x^{2}=0$ to get the answers $x=\frac{1}{2}(1 \pm \sqrt{5})$.