Using properties of determinants. Prove that`|(alpha,alpha^2,beta+gamma),(beta,beta^2,gamma+alpha),(gamma,gamma^2,alpha+beta)|=(beta-gamma)(gamma-alpha)(alpha-beta)(alpha+beta+gamma)`

Explanation

To prove that

\[
\left| \begin{array}{ccc}
\alpha & \alpha^2 & \beta + \gamma \\
\beta & \beta^2 & \gamma + \alpha \\
\gamma & \gamma^2 & \alpha + \beta
\end{array} \right| = (\beta - \gamma)(\gamma - \alpha)(\alpha - \beta)(\alpha + \beta + \gamma),
...