jmc

Since $g(x)=\sqrt[3]{\frac{x+3}{4}}$, we know that $g(2x)=\sqrt[3]{\frac{2x+3}{4}}$. Similarly, we see that $2(g(x))=2\sqrt[3]{\frac{x+3}{4}}$. This gives us the equation $$\begin{array}{rl}\sqrt[3]{\frac{2x+3}{4}}& =2\sqrt[3]{\frac{x+3}{4}}\\ \Rightarrow {\left(\sqrt[3]{\frac{2x+3}{4}}\right)}^3& ={\left(2\sqrt[3]{\frac{x+3}{4}}\right)}^3\\ \Rightarrow \frac{2x+3}{4}& =\frac{8(x+3)}{4}\\ \Rightarrow \frac{2x+3}{4}& =\frac{8x+24}{4}\\ \Rightarrow 2x+3& =8x+24\\ \Rightarrow -6x& =21\\ \Rightarrow x& =\boxed{-\frac{7}{2}}\end{array}$$