Δ = $\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{vmatrix}$ Applying $R_3$ → $R_3+R_1$ Δ = = α + β + γ $\begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ 1 & 1 & 1 \end{vmatrix}$ Applying $C_1 \rightarrow C_1 - C_2$ and $C_2 \rightarrow C_2 - C_3$ Δ = α + β + γ $\begin{vmatrix} \alpha-\beta & \beta-\gamma & \gamma \\ \alpha^2-\beta^2 & \beta^2-\gamma^2 & \gamma^2 \\ 0 & 0 & 1 \end{vmatrix}$ = α + β + γ (α - β) (β - γ) $\begin{vmatrix} 1 & 1 & \gamma \\ \alpha\beta & \beta+\gamma & \gamma^2 \\ 0 & 0 & 1 \end{vmatrix}$ = α+ β + γ (α - β) (β - γ) [1 (β + γ) - 1 (α + β)] = (α - β) (β - γ) (α + β + γ) (+ γ - α - β) = (α - β) (β - γ) (γ - α) (α + β + γ) Hence proved.