Euclidean Algorithm for GCD of polynomials I am struggling to use the Euclidean algorithm for polynomials.
Given something like $$GCD(x^5+1, x^3+1)$$
I can easily use it as follows: 
$$x^5+1 = x^2(x^3+1) -x^2 +1 \\
x^3+1 = -x(-x^2+1) + x +1 \\
-x^2+1 = (x+1)(-x+1)$$
GCD = x+1
But for something like $$GCD(2x^2+6x+3, 2x+1)$$ 
I cannot figure out how to do it using the same method. I start like this:
$$2x^2+6x+3 = x(2x+1) + 5x+3\\
2x+1 = \frac{2}{5}(5x+3) - \frac{1}{5}\\$$
and I am not sure how to continue. Any help would be appreciated, sorry for my poor attempt at using LaTeX.

As $(a,b)=(a+n\cdot b,b),$ where $n$ is any integer
$$(2x^2+6x+3,2x+1=((2x+1)x+5x+3,2x+1)=(5x+3,2x+1)$$
Now, $2(5x+3)-5(2x+1)=1\implies(5x+3,2x+1)=1$
