The remainder theorem states :

When we divide a polynomial #f(x)# by #(x-c)#, we get

#f(x)=(x-c)q(x)+r(x)#

and

#f(c)=0*q(x)+r=r#

We apply this theorem

#f(x)=(x^3+2x^2-3x+9#

Therefore,

#f(-3)=((-3)^3+2(-3)^2-3(-3)+9#

#=-27+18+9+9=9#

The remainder is #=9#

Let's perform the synthetic division to confirm the results

#color(white)(aa)##-3##color(white)(aaaaa)##|##color(white)(aaa)##1##color(white)(aaaaa)##2##color(white)(aaaaaa)##-3##color(white)(aaaaa)##9#

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

#color(white)(aaaa)##color(white)(aaaaaa)##|##color(white)(aaaa)##color(white)(aaa)##-3##color(white)(aaaaaaaa)##3##color(white)(aaaaa)##0#

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

#color(white)(aaaa)##color(white)(aaaaaa)##|##color(white)(aaa)##1##color(white)(aaaa)##-1##color(white)(aaaaaaa)##0##color(white)(aaaaa)##color(red)(9)#

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

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