jmc

By the Rational Root Theorem, any rational root $p/q$ of $g(x)$ must have $p$ divide $4$ and $q$ divide 1. In particular, this means any rational root must be an integer divisor of 4.

By trying out the integer factors of 4, we find that $g(2)=8-9\cdot 4+16\cdot 2-4=0$. Hence by the Factor theorem, $x-2$ is a factor of $g(x)$. With polynomial division, we can write $g(x)=(x-2)(x^2-7x+2).$ We can find the remaining roots of $g(x)$ by finding the roots of $x^2-7x+2$ using the quadratic formula. This gives us $$x=\frac{7\pm \sqrt{49-8}}{2}=\frac{7\pm \sqrt{41}}{2}.$$Since these are definitely not rational, the sum of the rational roots of $g(x)$ is $\boxed{2}.$