If $x=-1$ is a root of the equation $\left|\begin{array}{ccc}2-x& 3& 3\\ 3& 4-x& 5\\ 3& 5& 4-x\end{array}\right|=0$ then the other roots are

$c_3\to c_3-c_2$

$\Rightarrow \left|\begin{array}{ccc}2-x& 3& 0\\ 3& 4-x& 1+x\\ 3& 5& -x-1\end{array}\right|=0$

$R_3\to R_3+R_2$

$\Rightarrow \left|\begin{array}{ccc}2-x& 3& 0\\ 3& 4-x& 1+x\\ 6& 9-x& 0\end{array}\right|=0$

$\Rightarrow \left(1+x\right)\left[\left(2-x\right)\left(9-x\right)-18\right]=0$

$\Rightarrow \left(1+x\right)\left[18-11x+x^2-18\right]=0$

$\Rightarrow \left(1+x\right)\left(x^2-11x\right)=0$

$\Rightarrow x\left(1+x\right)\left(x-11\right)=0$

$\Rightarrow x=0\left(or\right)x=-1\left(or\right)x=11$