Solve the following quadratic equations by factorization:

We have been given

$\frac{x-1}{2 x+1}+\frac{2 x+1}{x-1}=\frac{5}{2}$

$2\left(x^{2}+1-2 x+4 x^{2}+1+4 x\right)=5\left(2 x^{2}-x-1\right)$

$10 x^{2}+4 x+4=10 x^{2}-5 x-5$

$9 x+9=0$

Therefore,

$9 x=-9$

$x=\frac{-9}{9}$

$x=-1$

Hence, $x=-1$.