Let's do a long division
#color(white)(aaaa)##12x^4##color(white)(aaaa)##12x^3##color(white)(aaaa)##-5x^2##color(white)(aaaa)##9x##color(white)(aaaa)##3##∣##x^2-4#
#color(white)(aaaa)##12x^4##color(white)(aaaaaaaa)####color(white)(aaaa)##-48x^2##color(white)(aaaa)####color(white)(aaaaaa)####∣##12x^2+12x+43#
#color(white)(aaaaaa)##0##color(white)(aaaaa)##12x^3##color(white)(aaa)####color(white)(aaaaaaaaaa)##9x##color(white)(aaaaaa)###
#color(white)(aaaaaa)####color(white)(aaaaaa)##12x^3##color(white)(aaaa)##+43x^2##color(white)(aa)##-48x##color(white)(aa)##+3#
#color(white)(aaaaaa)####color(white)(aaaaaa)##0##color(white)(aaaaaaa)##+43x^2##color(white)(aa)##+57x-172##color(white)(aaaaaa)###
#color(white)(aaaaaa)####color(white)(aaaaaa)####color(white)(aaaaaaaaaaaa)##0##color(white)(aa)##+57x+175##color(white)(aaaaaa)###
Therefore,
#(12x^4+12x^3-5x^2+9x+3)/(x^2-4)#
#=12x^2+12x+43+(57x+175)/(x^2-4)#