Let's perform the synthetic division

#color(white)(aaaa)##-3##color(white)(aaaaaa)##|##color(white)(aa)##1##color(white)(aaaa)##-2##color(white)(aaaaaa)##4##color(white)(aaaaaaa)##6#

#color(white)(aaaaaaaaaaaa)#*_**_**_**_**_*____

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaaa)##color(white)(aaaaa)##-3##color(white)(aaAaa)##15##color(white)(aaaaa)##-57#

#color(white)(aaaaaaaaaaaa)#*_**_**_**_**_*___

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaa)##1##color(white)(aaaaa)##-5##color(white)(aaaaaa)##19##color(white)(aaaa)##color(red)(-51)#

The remainder is #color(red)(-51)# and the quotient is #=x^2-5x+19#

Therefore,

#(x^3-2x^2+4x+6)/(x+3)=x^2-5x+19-51/(x+3)#