Correct Answer - Option 1 : (a - b), (b - c), (c - a), (a + b + c)
\(\left| {\begin{array}{*{20}{c}} a&{b + c}&{{a^2}}\\ b&{c + a}&{{b^2}}\\ c&{a + b}&{{c^2}} \end{array}} \right|\)
C1 → C1 + C2
\(=\left| {\begin{array}{*{20}{c}} {a + b + c}&{b + c}&{{a^2}}\\ {a + b + c}&{c + a}&{{b^2}}\\ {a + b + c}&{a + b}&{{c^2}} \end{array}} \right|\)
\(\Rightarrow {\left( {a + b + c} \right)\left| {\begin{array}{*{20}{c}} 1&{b + c}&{{a^2}}\\ 1&{c + a}&{{b^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
∴ (a + b + c) is one of the factor.
\(\Rightarrow \left( {a + b + c} \right){\left| {\begin{array}{*{20}{c}} 1&{b + c}&{{a^2}}\\ 1&{c + a}&{{b^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
R1 → R1 - R2 and R2 → R2 - R3
\(\Rightarrow\left( {a + b + c} \right) {\left| {\begin{array}{*{20}{c}} 0&{b - a}&{{a^2 - b^2}}\\ 0&{c - b}&{{b^2-c^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
R1 → R1 - R2
\(\Rightarrow\left( {a + b + c} \right){\left| {\begin{array}{*{20}{c}} 0&{c - a}&{{a^2 - c^2}}\\ 0&{c - b}&{{b^2-c^2}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
\(\Rightarrow\left( {a + b + c} \right) {\left( {c-a})( {b-c} \right)\left| {\begin{array}{*{20}{c}} 0&{1}&{{-(a + c)}}\\ 0&{-1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|}\)
R1 → R 1 + R2
\(\Rightarrow \left( {a + b + c} \right){\left( {c-a})( {b-c} \right)\left| {\begin{array}{*{20}{c}} 0&{0}&{{b-a}}\\ 0&{-1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
\(\Rightarrow \left( {a + b + c} \right){\left( {c-a})( {b-c})(a-b \right)\left| {\begin{array}{*{20}{c}} 0&{0}&{{-1}}\\ 0&{1}&{{b+c}}\\ 1&{a + b}&{{c^2}} \end{array}} \right|} \)
∴ (a - b), (b - c), (c - a), (a + b + c) are the factors.