Let $f(x)=2x^4 +3x^3 −19x^2 −28x+ 6$ and $g(x)=x^3 + 2x^2 -9x -18$ be polynomials in $\mathbb Q[x]$. Use the Euclidean Algorithm to determine the gcd in $\mathbb Q[x]$.
So far, I have the following:
$2x^4 + 3x^3 -19x^2 -28x +6 = 2x(x^3 +2x^2 -9x -18)+(-x^3 -x^2 +8x +6)$
$x^3 +2x^2 -9x -18 = 1(-x^3 -x^2 +8x +6)+(3x^2 -17x -24)$
$-x^3 -x^2 +8x +6 = [(-1/3)x](3x^2 -17x -24)+[(-20/3)x^2 +6]$
This is where I am stuck…
I'm not even sure if I am doing each step right. A little point in the right direction would be greatly appreciated. Thanks.