Eigenvalues of Matrix with variables Im supposed to find eigenvalues of following matrix:
$\begin{bmatrix}1 & -x^2+x-2 & 0\\1& 2x-1 & x-1\\ 0& 4 & 1\end{bmatrix}$
My attempt was to solve it the "normal way" to solve det(A-$\lambda$I) = 0
But since there is a variable in the matrix, this leads to a big cubic equation which i cannot solve easily, which looks like:
$$\lambda^3 - \lambda ^2 -2x\lambda + 5\lambda + x^2 \lambda -x\lambda +3x - x^2 -5 =0$$
I know that this equation will add up for $\lambda =1 $ but other than that i have no idea
am I missing something? is there a different way to attempt this?
Thanks

Since you know that $ \lambda = 1$ works, note that your equation factors as
$$(\lambda - 1) (x^2 - 3x + 5 + \lambda^2) = 0.$$
This gives that 
$$\lambda = 1, \lambda = \pm \sqrt{3x - x^2 - 5}.$$
