Show ring ideal is not principal ideal

[AcidRainLiTE]

Homework Statement

Show that the ideal
$$(3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x]$$
is not principal. (The parentheses mean 'the ideal generated by the elements enclosed in parentheses')

2. The attempt at a solution
I came up with a solution (see attachment), it is just rather convoluted. I feel like I am missing a more informative proof. I would like to take away as much as I can from this problem, so I am wondering if someone else sees a more insightful way to prove it. A brief summary of my solution is (see attachment for details):

(1) If it (call it I) is a principal ideal, then the fact that it contains 3 implies
$$I = a \mathbb{Z}[x] \text{ for some } a \in \mathbb{Z}$$

(2) But the fact that it contains $$x^3 - x^2 + 2x -1$$ implies a = 1 so
$$I = \mathbb{Z}[x]$$

(3) But $1 \notin I$ and hence $I \ne \mathbb{Z}[x]$. Contradiction. So I is not a principal ideal.

 

[AcidRainLiTE]

Forgot attachment