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)xcolor(white)(aaaa)-oocolor(white)(aaaa)-4color(white)(aaaa)2/3color(white)(aaaaaa)4color(white)(aaaa)+oo
color(white)(aaaa)3x-2color(white)(aaaa)-color(white)(aa)color(red)(∥)color(white)(a)-color(white)(aaaa)+color(white)(aaaa)+
color(white)(aaaa)3x-2color(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 [