Let's perform the synthetic division
color(white)(aa)-3color(white)(aaaaa)|color(white)(aaa)1color(white)(aaaaaaaa)0color(white)(aaaaaa)0color(white)(aaaaaaa)0color(white)(aaaaaa)9
color(white)(aaaaaaaaaaaa)------------
color(white)(aaaa)color(white)(aaaaaa)|color(white)(aaaaa)color(white)(aaaaa)-3color(white)(aaaaaa)9color(white)(aaaaa)-27color(white)(aaaaa)81
color(white)(aaaaaaaaaaaa)------------
color(white)(aaaa)color(white)(aaaaaa)|color(white)(aaa)1color(white)(aaaaaa)-3color(white)(aaaaaa)9color(white)(aaaaa)-27color(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)