For how many values (s) of $x$  in the closed interval $\left[-4,-1\right],$  is the matrix $\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|$  singular?

The given matrix is singular, if  $\left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|=0$ 

Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$

Applying $C_1\to C_1+C_2+C_3$

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

$\Rightarrow x^2\left(x+4\right)=0$

$\Rightarrow x=0,-4$

Hence, $x=-4$  is the only solution which lies in the given interval.