Solve the following quadratic equations by factorization:

We have been given

$\frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6}$

$6\left(x^{2}+1+2 x-x^{2}-1+2 x\right)=5\left(x^{2}-1\right)$

$5 x^{2}-24 x-5=0$

$5 x^{2}-25 x+x-5=0$

$5 x(x-5)+1(x-5)=0$

$(5 x+1)(x-5)=0$

Therefore,

$5 x+1=0$

$5 x=-1$

$x=\frac{-1}{5}$

or,

$x-5=0$

$x=5$

Hence, $x=\frac{-1}{5}$ or $x=5$.