A monic quadratic trinomial $P(x)$ is such that $P(P(P(x)))=0$ and $P(x)$ have a common root
A monic quadratic trinomial $P(x)$ is such that $P(P(P(x)))=0$ and $P(x)$ have a common root, then
$(A) P(0)\cdot P(1)>0$
$(B) P(0)\cdot P(1)<0$
$(C) P(0)\cdot P(1)=0$
Let $P(x)=x^2+bx+c$. I don't know how to do it further.
$$P(x)=x^2+bx+c\implies P(0)=c.$$ Let the common root be $\alpha$. We have, $$P(\alpha)=0,\; P(P(P(\alpha))=0 \implies P(P(0))=0\implies P(c)=0.$$ Since $c$ is a root of $P(x)$ and the product of the roots is $c$, the second root of $P(x)$ is $1$. $$\therefore\; P(0)P(1)=0$$