SOLUTION: Divide using long division x4 - 8x3 + 2x2 + 24x + 9 by x2 - 2x - 1

Divide using long division 
x4 - 8x3 + 2x2 + 24x + 9 by x2 - 2x - 1


Start with this:

                       
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9

x² goes into x4 x² times, so write that
above over the x² term

                         x²   
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9

Now multiply the x² by each of the terms at the far left,
and write the three terms that you get under the first three 
terms, and draw a line under them: 

            
                         x²          
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
              

Now we subtract vertically by mentally changing 
the sign of the bottom number and adding: 

                         x²   
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² 

Bring down the next term:

                         x²   
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x


Now x² goes into -6x³ -6x times, so we write
-6x³ ar the top:

                         x² -  6x   
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x

Now multiply the -6x by each of the terms at the far left,
and write the three terms that you get under the three 
terms at the bottom, and draw a line under them:

                         x² -  6x   
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x
                -6x² + 12x² +  6x
                -----------------
                       -9x² + 18x                     

Bring down the last term:
                         x² -  6x - 9  
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x
                -6x² + 12x² +  6x
                -----------------
                       -9x² + 18x + 9 

x² goes into -9x² -9 times, so write -9 on the
end at the top:

                         x² -  6x - 9  
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x
                -6x² + 12x² +  6x
                -----------------
                       -9x² + 18x + 9 

Multiply the -9 be each term at the left as
write them at the bottom:

                         x² -  6x - 9  
           --------------------------    
x² - 2x - 1)x4 - 8x³ +  2x² + 24x + 9
            x4 - 2x³ -   x²
            --------------
                -6x³ +  3x² + 24x
                -6x² + 12x² +  6x
                -----------------
                       -9x² + 18x + 9 
                       -9x² + 18x + 9
                       --------------

Then when you subtract you get 0 remainder,
so the answer is

x² - 6x - 9

Edwin