jmc

We can split the expression $\left|\frac{12}{x}+3\right|=2$ into two separate cases. In the first case, we have $$\begin{array}{rl}\frac{12}{x}+3& =2\\ \Rightarrow \frac{12}{x}& =-1\\ \Rightarrow -x& =12\\ \Rightarrow x& =-12\end{array}$$In the second case, $$\begin{array}{rl}\frac{12}{x}+3& =-2\\ \Rightarrow \frac{12}{x}& =-5\\ \Rightarrow -5x& =12\\ \Rightarrow x& =-\frac{12}{5}\end{array}$$Since each case gave us one possible value of $x$, the product of all possible values of $x$ is equal to $\left(-\frac{12}{5}\right)(-12)=\boxed{\frac{144}{5}}$.