The number of values of $x$ in $\left[-4,-1\right]$, for which the matrix $\left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]$ is singular, is

For the matrix to be singular, its determinant value should be zero

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

$\Rightarrow 3\left(-2+x+2\right)-\left(x-1\right)\left[6-\left(x+2\right)\left(x+3\right)\right]+2\left(-3+x+3\right)=0$

$\Rightarrow 3x+\left(x-1\right)x\left(x+5\right)+2x=0$

$$\begin{gathered} \Rightarrow x\left[5+x^2+4x-5\right]=0 \\ \Rightarrow x=0,-4 \end{gathered}$$

But, $x=0$ is rejected

Hence, only one value in $\left[-4,-1\right]$