Introduction

Needed to know this for Rational Zero Theorem so here goes

Important

In synthetic division you must include all powers if some are missing.

Here the is no x²

f(x) = 3x⁴ - 7x³ - 9x +12 where f(5)

So write this like this

f(x) = 3x⁴ - 7x³ + 0x² - 9x +12 where f(5)

5 | 3 - 7   0  - 9 +   12
  | ↓  15  40  200 +  955 
  -----------------------
    3   8  40  191 =  967

Now how to do it in full.

So we start with

(x³ -2x² -5x + 6) /  (x-3)

So to begin we list the coefficients of the functions in a row

 | 1 - 2 - 5 +  6
 |
-----------------

Next we solve the x-3, i.e x -3 = 0, therefore

x = 3

So first we bring down the 1

3 | 1 - 2 - 5 +  6
  | ↓
  -----------------
    1

Then we multiply the 3 by the number brought down (1) and place it under the next to the next coefficient

3 | 1 - 2 - 5 + 6
  | ↓   3
  -----------------
    1

Add the coefficient to the result 3, i.e. -2 + 3 = 1

3 | 1 - 2 - 5 + 6
  | ↓   3
  -----------------
    1   1

Then we multiply the 3 by the number brought down (1)

3 | 1 - 2 - 5 + 6
  | ↓   3   3 
 -----------------
    1   1

Add the coefficient to the result 3, i.e. -5 + 3 = -2

3 | 1 - 2 - 5 + 6
  | ↓   3   3 
  -----------------
    1   1  -2

Then we multiply the 3 by the number brought down (-2)

3 | 1 - 2 - 5 + 6
  | ↓   3   3 - 6
 -----------------
   1   1  -2

Add the coefficient to the result 3, i.e. 6 - 6 = 0

3 | 1 - 2 - 5 + 6
  | ↓   3   3 - 6
  -----------------
   1   1  -2   0

If the you get a zero then

   x - 3 is a factor and x =3

So taking the last line and adding the power of x in line with the question we see

  -----------------
    1   1  -2   0
    1x³+1x -2 <= This is the answer