Easiest way to perform Euclid's division algorithm for polynomials Let's say I have the two polynomials $f(x) = x^3 + x + 1$ and $g(x) = x^2 + x$ over $\operatorname{GF}(2)$ and want to perform a polynomial division in $\operatorname{GF}(2)$. 
What's the easiest and most bullet proof way to find the quotient $q(x) = x + 1$ and the remainder $r(x)=1$  by hand?
The proposal by the german edition of Wikipedia is rather awkward.

$f$ corresponds to the binary number $1011$ and $g$ to $110$ if you identify $x$ with $2$. Appending a $0$ (rsp. multiplication by $2$) corresponds to multiplying with $x$ and $\oplus$ (exclusive or) is addition.
1011:110 = 11, i.e., the quotient is $x+1$
110 
--- 
 111
 110
 ---
   1, i.e., the remainder is 1

