Find the values of k for which $\displaystyle \frac{x}{(x+1)^2(x-k)}$ has one stationary value.

I attempted to differentiate the equation using the cover-up and quotient rule, but the further i go the longer and more messy the result becomes, it'll take too long to post all my workings here

First:

$\displaystyle y= \frac{x}{(x+1)^2(x-k)}= \frac{Ax+B}{(x+1)^2}+ \frac{C}{x-k}$

when x=k with (x-k) covered, $\displaystyle C=\frac{k}{(k+1)^2}$

$\displaystyle x=(Ax+B)(x-k)+(\frac{k}{(k+1)^2})(x^2+2x+1)$

then comparing coefficients

$\displaystyle A+\frac{k}{(k+1)^2}=0$

$\displaystyle A=-\frac{k}{(k+1)^2}$

$\displaystyle -kB+\frac{k}{(k+1)^2}=0$

$\displaystyle B=\frac{1}{(k+1)^2}$

Then i inserted these values into the equation the tried to differentiate it with the quotient rule. This is where i get stuck.

thanks