Let's perform the synthetic division
#color(white)(aaaa)##-3##|##color(white)(aaaa)##1##color(white)(aaaaa)##2##color(white)(aaaaaa)##-3##color(white)(aaaaaaa)##9#
#color(white)(aaaaaaa)####|##color(white)(aaaa)####color(white)(aaaa)##-3##color(white)(aaaaaaa)##3##color(white)(aaaaaaa)##0#
#color(white)(aaaaaaaaa)###_________________________________________________________##
#color(white)(aaaaaaa)####|##color(white)(aaaa)##1##color(white)(aaa)##-1##color(white)(aaaaaaa)##0##color(white)(aaaaaaa)##color(red)(9)#
The remainder is #9# and the quotient is #=h^2-h#
Therefore,
#(h^3+2h^2-3h+9)/(h+3)=(h^2-h)+9/(h+3)#