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

$x+(a+b+c)$

The correct answer is option (2) : $x+(a+b+c)$

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

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

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

$⇒ (x+a+b+c) $ is a factor of Δ.