$\text{Theo Viet: }$
$\left\{{{x_{1}+x_{2}=-m}\atop{x_{1}x_{2}=1}}\right.$
$a) x_{1}^{2}+x_{2}^{2}$
$=(x_{1}+x_{2})^{2}-2x_{1}x_{2}$
$=m^{2}-2$
$b) x_{1}^{3}+x_{2}^{3}$
$=(x_{1}+x_{2})(x_{1}^{2}+x_{2}-x_{1}x_{2})$
$=(x_{1}+x_{2})[(x_{1}+x_{2})^{2}-3x_{1}x_{2}]$
$=-m.(m^{2}-3)$
$=3m-m^{3}$