原式左=1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)
=3/(x^2+5x+4)

∴9x=x^2+5x+4
(x-2)^2=0
x=2
楼主应该打错了,是x^2+3x+2