Determined Determinants and Discriminants

Level pending

\(Let\quad \alpha ,\beta \quad be\quad the\quad roots\quad of\quad a{ x }^{ 2 }+bx+c=0.\quad Let\quad { S }_{ n }={ \alpha }^{ n }{ +\beta }^{ n }\quad for\quad n\ge 1.\\ Evaluate\quad the\quad determinant:\begin{vmatrix} 3 & 1+{ S }_{ 1 } & 1+{ S }_{ 2 } \\ 1+{ S }_{ 1 } & 1+{ S }_{ 2 } & 1+{ S }_{ 3 } \\ 1+{ S }_{ 2 } & 1+{ S }_{ 3 } & 1+{ S }_{ 4 } \end{vmatrix}\)

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