Let's rewrite the inequality as
#(x-4)/(x^2+2x)=(x-4)/(x(x+2))#
And
Let #f(x)=(x-4)/(x(x+2))#
The domain of #f(x)# is #D_f(x)=RR-{-2,0} #
Now, we can make the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-2##color(white)(aaaaaa)##0##color(white)(aaaaaa)##4##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x+2##color(white)(aaaa)##-##color(white)(aaa)##∥##color(white)(aa)##+##∥##color(white)(aa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##x##color(white)(aaaaaaa)##-##color(white)(aaa)##∥##color(white)(aa)##-##∥##color(white)(aa)##+##color(white)(aaa)##+#
#color(white)(aaaa)##x-4##color(white)(aaaa)##-##color(white)(aaa)##∥##color(white)(aa)##-##∥##color(white)(aa)##-##color(white)(aaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaa)##-##color(white)(aaa)##∥##color(white)(aa)##+##∥##color(white)(aa)##-##color(white)(aaa)##+#
Therefore,
#f(x)<=0# when #x in ] -oo,-2 [ uu ] 0, 4] #
or, #x<-2# or # 0< x <=4#