I'm asked to find, if possible, a non triangular matrix which has $p(x) = (2 + x) (-x) (1-x)$ as its characteristic polynomial.
The book doesn't describe any method to do this, and after a while thinking I couldn't remember any process. So I searched here for other people struggling with the same problem and I found about the companion matrix. I managed to solve the problem, finding the matrix $A = \begin{bmatrix}0 & 0 & 0\\ 1 & 0 & 2\\ 0 & 1 & -1\end{bmatrix}$.
But this doesn't fully satisfy me, because I was thinking I should be able to solve this problem without having to resort to other methods not described in the book... What am I missing?
Yes companion matrix solves the question.
But, as the 3 eigenvalues $-2,0,1$ are distinct, any matrix with these eigenvalues can be written in terms of conjugation:
$$M=P \ \text{diag}(-2,0,1) \ P^{-1}$$
where $P$ is any $3 \times 3$ invertible matrix.