$\begin{array}{l}
a){x^2} + {y^2} = {\left( {x + y} \right)^2} - 2xy = {a^2} - 2b\\
b){x^3} + {y^3} = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)\\
= a\left( {{a^2} - 2b - b} \right) = a\left( {{a^2} - 3b} \right)\\
c){x^5} + {y^5} = \left( {{x^2} + {y^2}} \right)\left( {{x^3} + {y^3}} \right) - {x^2}{y^3} - {x^3}{y^2}\\
= \left( {{x^2} + {y^2}} \right)\left( {{x^3} + {y^3}} \right) - {x^2}{y^2}\left( {x + y} \right)\\
= a\left( {{a^2} - 2b} \right)\left( {{a^2} - 3b} \right) - {b^2}a
\end{array}$
