\(\frac{x+2}{x-2}-\frac{1}{x}=\frac{2}{x^2-2x}\)
\(\frac{x+2}{x-2}-\frac{1}{x}=\frac{2}{x^2-2x};x\ne2;x\ne0\)
\(\Leftrightarrow\frac{x+2}{x-2}-\frac{1}{x}-\frac{2}{x^2-2x}=0\)
\(\Leftrightarrow\frac{x+2}{x-2}-\frac{1}{x}-\frac{2}{x\times\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x\times\left(x+2\right)-\left(x-2\right)-2}{x\times\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x\times\left(x+2\right)-x+2-2}{x\times\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x^2+2x-x}{x\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x^2+x}{x\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x\left(x+1\right)}{x\left(x-2\right)}=0\)
\(\Leftrightarrow\frac{x+1}{x-2}=0\Rightarrow x+1=0\)
\(\Rightarrow x=-1\)
