Let's perform the synthetic division

#color(white)(aa)##-3##color(white)(aaaaa)##|##color(white)(aaa)##1##color(white)(aaaaaaaa)##0##color(white)(aaaaaa)##0##color(white)(aaaaaaa)##0##color(white)(aaaaaa)##9#

#color(white)(aaaaaaaaaaaa)##------------#

#color(white)(aaaa)##color(white)(aaaaaa)##|##color(white)(aaaaa)##color(white)(aaaaa)##-3##color(white)(aaaaaa)##9##color(white)(aaaaa)##-27##color(white)(aaaaa)##81#

#color(white)(aaaaaaaaaaaa)##------------#

#color(white)(aaaa)##color(white)(aaaaaa)##|##color(white)(aaa)##1##color(white)(aaaaaa)##-3##color(white)(aaaaaa)##9##color(white)(aaaaa)##-27##color(white)(aaaaa)##color(red)(90)#

The remainder is #color(red)(90)# and the quotient is #=x^3-3x^2+9x-27#

#(x^4+9)/(x+3)=x^3-3x^2+9x-27+90/(x+3)#