Answer
$24-26\lambda+9\lambda^2-\lambda^3$
Work Step by Step
$A=\begin{bmatrix}
4-\lambda&0&0\\5&3-\lambda&2\\-2&0&2-\lambda
\end{bmatrix}$
$det(A)=(4-\lambda)\left|\begin{matrix}
3-\lambda&2\\0&2-\lambda
\end{matrix}\right|=(4-\lambda)(3-\lambda)(2-\lambda)=0$
$(12-7\lambda+\lambda^2)(2-\lambda)=24-26\lambda+9\lambda^2-\lambda^3$