The other tutor's answer is wrong.

Start with this:

i | 4  -9i   5   8-4i
  |__________________   
  
Bring down the 4

i | 4  -9i   5   8-4i
  |__________________   
    4  
 
Multiply the 4 by the i gatting 4i. Write 4i above and to the 
right of the 4 that you brought down, underneath the -9i:

i | 4  -9i   5   8-4i
  |     4i           
    4

Add the -9i and the 4i getting -5i.  Write that below the line
under the 4i

i | 4  -9i   5   8-4i
  |     4i           
    4  -5i 
   
Multiply the -5i by the i and get -5i², then
replace i² by -1 and get -5(-1) or 5 and write
5 above and to the right of the -5i under the
other 5:

i | 4  -9i   5   8-4i
  |     4i   5          
    4  -5i 

Add the 5 and the 5 getting 10 and
write that at the bottom under the
5's
   
i | 4 -9i    5   8-4i
  |    4i    5       
    4 -5i   10

Multiply the 10 by the i getting 10i.
Write that above and to the right of
the 10 underneath the -4i

i | 4 -9i    5    8- 4i
  |    4i    5      10i
    4 -5i   10

Add the 8-4i and the 10i getting 8+6i.
Write that at the bottom under the +10i
 
i | 4 -9i    5    8- 4i
  |    4i    5      10i
    4 -5i   10    8+ 6i

Use the first three terms on the bottom row 
as coefficients of a quadratic equation,
then put the last number on the bottom over
the original divisor x-i

4x² - 5ix + 10 + 


Edwin