Let's rewrite and factorise the inequality
#x^3+2x^2<=8x#
#x^3+2x^2-8x<=0#
#x(x^2+2x-8)<=0#
#x(x-2)(x+4)<=0#
Let #f(x)=x(x-2)(x+4)#
Now, we can build the sign chart
#color(white)(aaaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaa)##0##color(white)(aaaaa)##2##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x+4##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x##color(white)(aaaaaaaaa)##-##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-2##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(x)<=0#, when #x in ]-oo,-4]uu[0, 2]#