\(\frac{240}{x-12}-\frac{240}{x}=\frac{5}{3}\)
\(\Leftrightarrow\frac{240}{x-12}-\frac{240}{x}=\frac{5}{3}\left(x\ne12,x\ne0\right)\)
\(\Leftrightarrow\frac{240}{x-12}-\frac{240}{x}-\frac{5}{3}=0\)
\(\Leftrightarrow\frac{720x-720x+8640-5x^2+60x}{3x\left(x-12\right)}=0\)
\(\Leftrightarrow8640-5x^2+60x=0\)
\(\Leftrightarrow5\left(1728-x^2+12x\right)=0\)
\(\Leftrightarrow5\left(-x^2+12x+1728\right)=0\)
\(\Leftrightarrow5\left(-x^2+48x-36x+1728\right)=0\)
\(\Leftrightarrow5\left[-x\left(x-48\right)-36\left(x-48\right)\right]=0\)
\(\Leftrightarrow5\left[-\left(x-48\right)\right].\left(x+36\right)=0\)
\(\Leftrightarrow-5\left(x-48\right).\left(x+36\right)=0\)
\(\Leftrightarrow\left(x-48\right)\left(x+36\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x-48=0\\x+36=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=48\\x=-36\end{matrix}\right.,x\ne12,x\ne0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=48\\x=-36\end{matrix}\right.\)