Quotient and Remainder algorithm for Euclidean's

Question

I've done the Euclidean algorithm before, but I'm trying to come up with separate algorithms for the quotient and remainder part that I can implement in some pseudo-code.

Ex: I have two polynomials '$a$' and '$b$' in (for simplicity's sake) $GF(2)[x]$. Then I have two other polynomials '$q$' and '$r$' (quotient and remainder) such that $a=bq+r$.

What I want to do is come up with two algorithms: $Quot(a, b)$ and $Rem(a, b)$ which compute '$q$' and '$r$'.

The Euclidean algorithm itself isn't hard to implement. A simple implementation is:

def gcd(a, b):
     while b:
      a,b = b, a % b
    return a

Then solve backwards using r* = qr + (new r)

Just having trouble figuring this out. It seems like something very simple that I'm not getting.