Polynomials

We want $Q(x)$ to be a quadratic polynomial (so that $P(Q(x))$ has degree $6$ ). For $P(x)$ to divide $P(Q(x))$ , we want $P(Q(x))$ to have zeros at $3,4,5$ , and hence each of $Q(3),Q(4),Q(5)$ must belong to $\{3,4,5\}$ .

There is a unique polynomial $Q_{a,b,c}(x)$ of degree at most $2$ such that $Q_{a,b,c}(3)=a$ , $Q_{a,b,c}(4)=b$ and $Q_{a,b,c}(5) = c$ , and so each valid $Q(x)$ must be a quadratic $Q_{a,b,c}(x)$ where $a,b,c \in \{3,4,5\}$ . Considering the $27$ polynomials $Q_{a,b,c}(x)$ for $a,b,c \in \{3,4,5\}$ , the polynomial $Q_{a,b,c}(x)$ will have degree $2$ except when $a,b,c$ are in arithmetic progression (in which case $Q_{a,b,c}(x)$ will be linear or constant). Only the triples $(3,3,3)\,,\,(4,4,4)\,,\,(5,5,5)\,,\,(3,4,5)\,,\,(5,4,3)$ are in arithmetic progression, and so there are $\boxed{22}$ polynomials $Q(x)$ .