Let's perform the synthetic division
#color(white)(aaaa)##2##color(white)(aaaa)##|##color(white)(aaaa)##3##color(white)(aaaa)##9##color(white)(aaaaa)##-2##color(white)(aaaa)##2##color(white)(aaaaa)##-7#
#color(white)(aaaaaaaaaaaa)#_________
#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaaa)##color(white)(aaa)##6##color(white)(aaaaaa)##30##color(white)(aaa)##56##color(white)(aaaaa)##116#
#color(white)(aaaaaaaaaaaa)#________
#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaa)##3##color(white)(aaa)##15##color(white)(aaaaa)##28##color(white)(aaa)##58##color(white)(aaaaa)##color(red)(109)#
The remainder is #=109# and the quotient is #=3x^3+15x^2+28x+58#
Therefore,
#(3x^4+9x^3-2x^2+2x-7)/(x-2)=3x^3+15x^2+28x+58+(109)/(x-2)#