Start by factorising the numerator and the denominator
#12n^3+16n^2-3n-4=12n^3-3n+16n^2-4#
#=3n(4n^2-1)+4(4n^2-1)#
#=(4n^2-1)(3n+4)#
#=(2n+1)(2n-1)(3n+4)#
#8n^3+12n^2+10n+15=8n^3+10n+12n^2+15#
#=2n(4n^2+5)+3(4n^2+5)#
#=(4n^2+5)(2n+3)#
Finally,
#(12n^3+16n^2-3n-4)/(8n^3+12n^2+10n+15)=((2n+1)(2n-1)(3n+4))/((4n^2+5)(2n+3))#
Let #f(n)=((2n+1)(2n-1)(3n+4))/((4n^2+5)(2n+3))#
The term #(4n^2+5)>0#
We can construct the sign chart
#color(white)(aaaa)##n##color(white)(aaa)##-oo##color(white)(aaa)##-3/2##color(white)(aaa)##-4/3##color(white)(aaaa)##-1/2##color(white)(aaaa)##1/2##color(white)(aaaa)##+oo#
#color(white)(aaaa)##2n-3##color(white)(aaaa)##-##color(white)(a)##||##color(white)(aa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##3n+4##color(white)(aaaa)##-##color(white)(a)##||##color(white)(aa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##2n+1##color(white)(aaaa)##-##color(white)(a)##||##color(white)(aa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##2n-1##color(white)(aaaa)##-##color(white)(a)##||##color(white)(aa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(n)##color(white)(aaaaaa)##+##color(white)(a)##||##color(white)(aa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(n)>0# when #n in (-oo,-3/2)uu(-4/3,-1/2)uu(1/2,+oo)#
graph{(12x^3+16x^2-3x-4)/(8x^3+12x^2+10x+15) [-5.944, 3.92, -2.265, 2.665]}