Hello, KC!
\(\displaystyle (x^3\,+\,3x^2\,-\,10x\,-\,24)\:\div\
x\,-\,3)\)
Since explaining every step will require pages and pages of explanation,
\(\displaystyle \;\;\)I'll assume you know the procedures . . . and I'll just give you the results.
Long division:
. . . . . . . . . . . . . . \(\displaystyle x^2\;+\;6x\;+\;8\)
. . . . . . . . .- - - - - - - - - - - - - - - -
\(\displaystyle \;\;x\,-\,3\
\:x^3\,+\,3x^2\,-10x\,-\,24\)
. . . . . . . . . \(\displaystyle x^3\,-\,3x^2\)
. . . . . . . . .- - - - - - -
. . . . . . . . . . . . . .\(\displaystyle 6x^2\,-\,10x\)
. . . . . . . . . . . . . .\(\displaystyle 6x^2\,-\,18x\)
. . . . . . . . . . . . . .- - - - - - -
. . . . . . . . . . . . . . . . . . . \(\displaystyle 8x\,-\,24\)
. . . . . . . . . . . . . . . . . . . \(\displaystyle 8x\,-\,24\)
. . . . . . . . . . . . . . . . . . . - - - - - - -
Synthetic division:
. . . \(\displaystyle 3\;|\;1\quad3\quad\)-\(\displaystyle 10\quad\)-\(\displaystyle 24\)
. . . . . \(\displaystyle |\;\;\:3\quad\,18\quad\:24\)
. . . . . - - - - - - - - - -
. . . . . . \(\displaystyle 1\quad6\qquad\;\8\qquad\;0\)
.