$\begin{align}
& {{x}^{2}}-x-1=0\to S=1,P=-1 \\
& \left( {{\alpha }^{3}}+{{\beta }^{3}} \right)\left( {{\alpha }^{4}}+{{\beta }^{4}} \right)={{\alpha }^{7}}+{{\beta }^{7}}+{{\alpha }^{3}}{{\beta }^{4}}+{{\alpha }^{4}}{{\beta }^{3}} \\
& \to {{\alpha }^{7}}+{{\beta }^{7}}=\left( {{\alpha }^{3}}+{{\beta }^{3}} \right)\left( {{\alpha }^{4}}+{{\beta }^{4}} \right)-{{\alpha }^{3}}{{\beta }^{4}}-{{\alpha }^{4}}{{\beta }^{3}}=\left( {{S}^{3}}-3PS \right)\left( {{\left( {{S}^{2}}-2P \right)}^{2}}-2{{P}^{2}} \right)-S{{P}^{3}} \\
& =\left( {{\left( 1 \right)}^{3}}-3\left( -1 \right)\left( 1 \right) \right)\left( {{\left( {{\left( 1 \right)}^{2}}-2\left( -1 \right) \right)}^{2}}-2{{\left( -1 \right)}^{2}} \right)-\left( 1 \right){{\left( -1 \right)}^{3}}=29 \\
\end{align}$