abstract algebra - Greatest common divisor of polynomials over $mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have guessed their greatest common divisor: $(x-1)$, but I did it by looking for roots of both polynomials, and now I am stuck. How do I find the greatest common divisor using the Euclid algorithm? I started with $f(x) = g(x) + 3(2x^2 - x - 1)$, but then things go nuts, and I can't use Bézout's identity to bring it all back to $gcd(f,g) = a \cdot f + b \cdot v$.