The values of  $\text{'}x\text{'}$  for which the matrix  is $\left[\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right]$  non-singular are   

$\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|$

$C_1\to C_1+C_2+C_3$ 

$(x+a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& x+b& c\\ 1& b& x+c\end{array}\right|$

$\begin{array}{c}{R}_2\to R_2-R_1\\ R_3\to R_3-R_1\end{array}$

$(x+a+b+c)\left|\begin{array}{ccc}1& b& c\\ 0& x& 0\\ 0& 0& x\end{array}\right|=x^2(x+a+b+c)$

Given matrix is  non singular

$\Rightarrow x^2\left(x+a+b+c\right)\ne 0\Rightarrow x\ne 0,x\ne -a-b-c$