jmc

Use the Euclidean algorithm on $f(x)$ and $g(x)$. $$\begin{array}{rl}h(x)& =\gcd (f(x),g(x))\\ & =\gcd (12x+7,5x+2)\\ & =\gcd (5x+2,(12x+7)-2(5x+2))\\ & =\gcd (5x+2,2x+3)\\ & =\gcd (2x+3,(5x+2)-2(2x+3))\\ & =\gcd (2x+3,x-4)\\ & =\gcd (x-4,(2x+3)-2(x-4))\\ & =\gcd (x-4,11)\end{array}$$From applying the Euclidean algorithm, we have that the greatest common divisor of $f(x)$ and $g(x)$ is 11 if and only if $x-4$ is a multiple of 11. For example, note that $f(4)=55$ and $g(4)=22$, and the greatest common divisor of 55 and 22 turns out to be 11. If $x-4$ is not a multiple of 11, then the greatest common divisor of $f(x)$ and $g(x)$ must be one, since 11 is prime and therefore has no other factors. It follows that $h(x)$ can take on two distinct values; 1 and 11. The sum of all possible values of $h(x)$ is therefore $1+11=\boxed{12}$.