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)##x##color(white)(aaaa)##-oo##color(white)(aaaaa)##-3##color(white)(aaaaaaa)##4##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x+3##color(white)(aaaaaa)##-##color(white)(aa)##0##color(white)(aaa)##+##color(white)(aaaaaa)##+#
#color(white)(aaaa)##x-4##color(white)(aaaaaa)##-##color(white)(aa)####color(white)(aaaa)##-##color(white)(aa)##0##color(white)(aaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##+##color(white)(aa)##0##color(white)(aaa)##-##color(white)(aa)##0##color(white)(aaa)##+#
Therefore,
#f(x)<=0# when # x in [-3,4] #
graph{(x^2-x-12)/(x^2+4) [-10, 10, -5, 5]}