\(\dfrac{3x+m}{3x-m}-\dfrac{6}{m+3x}=\dfrac{9x^2}{9x^3-m^2}\)
\(\Leftrightarrow\dfrac{\left(3x+m\right)\left(3x+m\right)-6\left(3x-m\right)}{\left(3x-m\right)\left(3x+m\right)}=\dfrac{9x^2}{\left(3x-m\right)\left(3x+m\right)}\)
\(\Leftrightarrow\left(3x+m\right)^2-6\left(3x-m\right)=9x^2\)
\(\Leftrightarrow9x^2+6mx+m^2-18x+6m=9x^2\)
\(\Leftrightarrow6mx+m^2-18x+6m=0\)
\(x=\dfrac{8}{3}\)
\(\Leftrightarrow6.\dfrac{8}{3}m+m^2-18.\dfrac{8}{3}+6m=0\)
\(\Leftrightarrow16m+m^2-48+6m=0\)
\(\Leftrightarrow m^2+22m-48=0\)
\(\Leftrightarrow\left(m-2\right)\left(m+24\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}m=2\\m=-24\end{matrix}\right.\)
