Let #f(x)=((3x-2)(x-4))/(x+4)^2#
The domain of #f(x)# is #D_f(x)=RR-{-4}#
The denominator is #>0, AA x in D_f(x)#
Let's do the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaa)##2/3##color(white)(aaaaaa)##4##color(white)(aaaa)##+oo#
#color(white)(aaaa)##3x-2##color(white)(aaaa)##-##color(white)(aa)##color(red)(∥)##color(white)(a)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##3x-2##color(white)(aaaa)##-##color(white)(aa)##color(red)(∥)##color(white)(a)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##+##color(white)(aa)##color(red)(∥)##color(white)(a)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(x)<0#, when #x in ] 2/3 , 4 [ #
