I am trying to find all values for $\alpha$ and $\beta$ for which 

$$ A(\alpha, \beta)= 
\left[
\begin{matrix}
3&0&-2\\\alpha&3&2\\-2&2&\beta 
\end{matrix}
\right]
$$

is singular.

My understanding is that I need to find values $\beta$ and $\alpha$ for which $det(A)=0$

Therefore so,

$$\det(A) = 3
\begin{bmatrix}
3&2\\2&\beta
\end{bmatrix} - 0 + (-2)\left[
\begin{matrix}
\alpha&3\\-2&2
\end{matrix}
\right]\\= 3(3\beta - 4) - 2(2\alpha +6) \\ = 9\beta -12 -4\alpha -12$$

If my workings are correct, then what is $\beta,\alpha?$