How to calculate this determinant (Finding characteristic polynomial).

Question

I am trying to calculate this determinant in order to find a characteristic polynomial of a matrix, however I couldn't see any operation that I can do to make my life easier calculating this, so I tried to calculate the determinant according to column $3$ since it has a zero, but it was such a long journey and in the end I got a polynomial with degree of $3$, so I also needed to find its roots (eigen values) once again and it took alot of time.
So I'm wondering if there's any easy way to calculate this determinant since most likely I just couldn't see the right operation to do to calculate it and took the long road.
Here's the determinant: \begin{vmatrix} x-2 & -3 & -2 \\ 1 & x-3 & 0 \\ -1 & -1 & x+1\end{vmatrix}
Any feedback and tips are really appreciated, thanks in advance.

Answer 1

Hint: There a simple method, you have just to do: \begin{align} (x-2)\begin{vmatrix} x-3 & 0 \\ -1 & x+1 \end{vmatrix} + (-1)(-3)\begin{vmatrix} 1 & 0 \\ -1 & x+1 \end{vmatrix} + (-2)\begin{vmatrix} 1 & x-3 \\ -1 & -1 \end{vmatrix} \end{align}

Can you continue from here?