x/(x+1)-(x+1)/x=k/(x^2+x)
[x^2-(x+1)^2]/(x^2+x)=k/(x^2+x)
x^2-(x+1)^2=k,且x^2+x≠0,即x≠0且x≠-1
-2x-1=k
x=-(k+1)/2
有增根就是-(k+1)/2=0或-(k+1)/2=-1
k=-1或k=1