Factorise the numerator
x^2-x-12=(x+3)(x-4)
Therefore,
(x^2-x-12)/(x^2+4)<=0
<=>, ((x+3)(x-4))/(x^2+4)<=0
AA x in RR, (x^2+4)>0
Let f(x)=((x+3)(x-4))/(x^2+4)
Let's build the sign chart
color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaa)-3color(white)(aaaaaaa)4color(white)(aaaaa)+oo
color(white)(aaaa)x+3color(white)(aaaaaa)-color(white)(aa)0color(white)(aaa)+color(white)(aaaaaa)+
color(white)(aaaa)x-4color(white)(aaaaaa)-color(white)(aa)#color(white)(aaaa)-#color(white)(aa)0color(white)(aaa)+
color(white)(aaaa)f(x)color(white)(aaaaaaa)+color(white)(aa)0color(white)(aaa)-color(white)(aa)0color(white)(aaa)+
Therefore,
f(x)<=0 when x in [-3,4]
graph{(x^2-x-12)/(x^2+4) [-10, 10, -5, 5]}