jmc

We make the substitution $y={\left(\frac{x}{x+1}\right)}^2$ to simplify the equation, so that $$\frac{y+11}{y+1}=2.$$Multiplying by $y+1$ gives $y+11=2y+2,$ so $y=9.$ Therefore, we have $$\frac{x}{x+1}=\pm 3.$$Then, either $x=3(x+1)$ or $x=-3(x+1).$ These give solutions $x=\boxed{-\frac{3}{2}}$ and $x=\boxed{-\frac{3}{4}},$ respectively.