linear algebra - How do I find the characteristic polynomial and eigenvalues?

For the following matrix, compute

1. its characteristic polynomial


2. its eigenvalues

$$A = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & -5 & 4\end{bmatrix}$$

So I think I know to find the characteristic polynomial, I have to compute det(A-$\lambda$I) = 0.
Which gives...

$$A-\lambda I = \begin{bmatrix}-\lambda & 1 & 0 \\ 0 & -\lambda & 1 \\ 2 & -5 & 4-\lambda\end{bmatrix}$$

$$-\lambda\begin{vmatrix}-\lambda & 1 \\ -5 & 4-\lambda\end{vmatrix} - 1 \begin{vmatrix}0 & 1 \\2 & 4-\lambda\end{vmatrix}$$

After calculating the determinant I get, $-\lambda(-4\lambda + \lambda^{2}+5) +2 = 0$

What do I do next? I'm completely stuck.