Find a content of $(1 + w)x^4 + ( -1 + 2w )x^3 + (1-2w )x^2 + 3x + (2 + 3w ) \in \mathbb{Z}[w][x]$
My attempt:
I see that $w+1$ is a unit, $( -1 + 2w )$ and $(1-2w )$ are associates, $3=(1+2w)(1+2w^2)$, and I know the list of units in $\mathbb{Z}[w]$ is $\{\pm 1, \pm w ,\pm w^2\}$ ,where $w^2=-w-1$,
So, how do I proceed now? Since $w=\frac{-1+\sqrt 3 i}{2}$ dividing coefficients with each other doesn't help as much as it does in case of $\mathbb Z[i]$
As @random suggests, I will compare the norms of the coefficients, where the norm function is $N(a+bw)=a^2-ab+b^2$
$N(w+1)=1$, since $w+1$ is a unit,
$N(-1+2w)=N(1-2w)=7$,
$N(3)=9$,
$N(2+3w)=7$,
So, a content of the polynomial could be $1\in \mathbb{Z}[w]$. Am I right?