$Δ=\begin{vmatrix}a+x&b&c\\b&x+c&a\\c&a&x+b\end{vmatrix}$. Which of the following is a factor for the above determinant?

$x + (a+b+c)$

We have,

$Δ=\begin{vmatrix}a+x&b&c\\b&x+c&a\\c&a&x+b\end{vmatrix}$

$⇒Δ=\begin{vmatrix}x+a+b+c&b&c\\x+a+b+c&x+c&a\\x+a+b+c&a&x+b\end{vmatrix}$ Applying $C_1→C_1+C_2+C_3$

$⇒Δ=(x+a+b+c)\begin{vmatrix}1&b&c\\1&x+c&a\\1&a&x+b\end{vmatrix}$

$⇒(x+a+b+c)$ is a factor of Δ.