Which of the following is a factor of the determinant […] ?

The correct answer is Option (2) → \(a+b+c\)

Each row contains a, b and c once. So the column operation \(C_1\rightarrow C_1+C_2+C_3\) makes every entry of the first column equal to a + b + c:

    \(\begin{vmatrix}a+b+c & b & c\\ a+b+c & c & a\\ a+b+c & a & b\end{vmatrix}=(a+b+c)\begin{vmatrix}1 & b & c\\ 1 & c & a\\ 1 & a & b\end{vmatrix}\)

So a + b + c is a factor. (In full, the determinant equals \(-(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\).)

The other options are not factors. A factor must make the determinant zero whenever the factor is zero. With a = 1, b = −1, c = 1 we have a + b = 0, but the determinant is \(-(1)(4)=-4\). In the same way b + c = 0 at (a, b, c) = (1, 1, −1) and a − b + c = 0 at (1, 2, 1), yet the determinant is −4 at both.