Let $$f(x) = a_0 + a_1x + a_2x^2 + a_3x^3+a_4x^4\\ g(x) = b_0 + b_1x + b_2x^2 + b_3x^3+b_4x^4 \\ h(x) = c_0 + c_1x + c_2x^2 + c_3x^3+c_4x^4\\ \text{and }\; (x-\alpha)(x-\beta)(x-\gamma) = x^3+px^2+qx+r$$
Show that $$ \begin{vmatrix} f(\alpha)&f(\beta)&f(\gamma)\\ g(\alpha)&g(\beta)&g(\gamma)\\ h(\alpha)&h(\beta)&h(\gamma) \end{vmatrix} = \begin{vmatrix} 1&\alpha&\alpha^2\\1&\beta&\beta^2\\1&\gamma&\gamma^2 \end{vmatrix} .\begin{vmatrix} a_0&a_1&a_2&a_3&a_4\\ b_0&b_1&b_2&b_3&b_4\\c_0&c_1&c_2&c_3&c_4\\r&p&q&1&0\\0&r&q&p&1\end{vmatrix}$$ This question in the book : "Problems in linear algebra" by I. V. Proskuryakov (1978). Their hint is express both side in to sum of some determinants, then showw that they are equals. Help me please! Thanks alot! (Sorry about my bad English, that also why i hesitate and can't provide more information for you guy).