jmc

In $f(x)$, all terms will have a multiple of $x$ except for the constant term, which is the multiple of the four constants $4,1,6$, and $9$.

Recall (from the Euclidean algorithm) that the greatest common divisor of $a$ and $b$ is the same as the greatest common divisor of $a$ and $a-kb$ where $k,a,$ and $b$ are any integers. Therefore, finding the greatest common divisor of $f(x)$ and $x$ is the same as finding the greatest common divisor of $x$ and the constant term of $f(x)$. Therefore, we want to find $$\begin{array}{rl}\text{gcd}((3x+4)(7x+1)(13x+6)(2x+9),x)& =\text{gcd}(4\cdot 1\cdot 6\cdot 9,x)\\ & =\text{gcd}(216,x)\end{array}$$Since $15336$ is a multiple of $216$, the greatest common divisor of $f(x)$ and $x$ is $\boxed{216}$.