\(\frac{1}{x+1}-\frac{5}{x-1}=\frac{15}{\left(x+1\right)\cdot\left(2-x\right)}\)
\(\Leftrightarrow\frac{x-1}{x^2-1}-\frac{5x+5}{x^2-1}=\frac{15}{\left(x+1\right)\cdot\left(2-x\right)}\)
\(\Leftrightarrow\frac{x-1-5x-5}{x^2-1}=\frac{15}{\left(x+1\right)\cdot\left(2-x\right)}\)
\(\Leftrightarrow\frac{-4x-6}{x^2-1}=\frac{15}{\left(x+1\right)\cdot\left(2-x\right)}\)
\(\Leftrightarrow\frac{4x^2-2x-12}{\left(x^2-1\right)\cdot\left(2-x\right)}=\frac{15x-15}{\left(x^2-1\right)\cdot\left(2-x\right)}\)
\(\Leftrightarrow\frac{4x^2-2x-12-15x+15}{\left(x^2-1\right)\cdot\left(2-x\right)}=0\)
\(\Leftrightarrow\frac{4x^2-17x+3}{\left(x^2-1\right)\cdot\left(2-x\right)}=0\)
\(\Rightarrow4x^2-17x+3=0\)
