We solve this inequality with a sign chart
Let #f(x)=((x+3)(x+5))/(x+2)#
The domain of #f(x)# is #D_f(x)=RR-{-2}#
Let build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-5##color(white)(aaaa)##-3##color(white)(aaaaaa)##-2##color(white)(aaaaaaa)##+oo#
#color(white)(aaaa)##x+5##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaaa)##+#
#color(white)(aaaa)##x+3##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+##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)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##||##color(white)(aaaa)##+#
Therefore,
#f(x)>=0# when # x in [-5,-3] uu]-2, +oo[#