You can find the remainder right away by simply plugging in [tex]x=3[/tex]. The polynomial remainder theorem guarantees that the value of [tex]p(3)[/tex] is the remainder upon dividing [tex]p(x)[/tex] by [tex]x-3[/tex], but I digress...
Synthetic division yields
3 | 2 -11 18 -15
. | 6 -15 9
- - - - - - - - - - - - - - - - -
. | 2 -5 3 -6
which translates to
[tex]\dfrac{2x^3-11x^2+18x-15}{x-3}=2x^2-5x+3-\dfrac6{x-3}[/tex]
(and note that [tex]p(3)=2(3)^3-11(3)^2+18(3)-15=-6[/tex], as expected)
