[originally on newsgroup; here, with all corrections, plus $deconv$]

let's look at an example computed by hand: 
 
    6 x^4 - 3 x^3 + 17 x^2 - 7x + 9 divided by 3 x^2 + 1
 
let's write 3 x^2 + 1 as 3 x^2 + 0 x + 1.
 
 
                                    2 x^2  - 1 x + 5, with remainder -6 x + 4
                 +---------------------------------
 3 x^2 + 0 x + 1 | 6 x^4 - 3 x^3 + 17 x^2 - 7 x + 9   
                   6 x^4 + 0 x^3 +  2 x^2             2 x^2 = 6 x^4 / 3 x^2
                   --------------------------------
                         - 3 x^3 + 15 x^2 - 7 x + 9  
                         - 3 x^3 +  0 x^2 - 1 x      -1 x   = - 3 x^3 / 3 x^2
                         --------------------------
                                   15 x^2 - 6 x + 9   
                                   15 x^2 + 0 x + 5   5     = 15 x^2 / 3 x^2
                                   ----------------
                                          - 6 x + 4

conceptually, what is going on?

1. shift divisor 3 x^2 + 0 x + 1 until it's lead term 3 x^2 is lined up 
   with the lead term 6 x^4 of dividend 6 x^4 - 3 x^3 + 17 x^2 - 7x + 9 

2. divide dividend's remaining lead term (e.g. 6 x^4) by divisor's lead
   term (3 x^2) to get next term (e.g. 6 x^4 / 3 x^2 = 2 x^2) in quotient

3. subtract off corresponding multiple (e.g. 2 x^2 * (3 x^2 + 0 x + 1))
   of divisor from dividend

4. repeat until left with remainder
_______________________________________________________________________________

let's check our answer using $deconv$:

p1 = [9 -7 17 -3 6]; % canonical rep. of 6 x^4 - 3 x^3 + 17 x^2 - 7x + 9
p2 = [1 0 3]; % canonical rep. of 3 x^2 + 1
[q,r] = deconv(fliplr(p1), fliplr(p2)); % compute quotient and remainder
q = fliplr(q); r = fliplr(r); % fliplr here and above is because matlab's
                              % poly rep is left-to-right instead of 
                              % left-to-right
q % should be [5 -1 2], the canonical rep. of 2 x^2  - 1 x + 5
r % should be (something like) [4 -6], the canonical rep. of -6 x + 4