Polynom/Einsetzung/3/Aufgabe/Lösung

Es ist

${}{\begin{aligned}{\left(-5X^{3}-X^{2}+{\sqrt {2}}X+{\sqrt {5}}\right)}{\left({\sqrt {2}}+{\sqrt {3}}\right)}&=-5{\left({\sqrt {2}}+{\sqrt {3}}\right)}^{3}-{\left({\sqrt {2}}+{\sqrt {3}}\right)}^{2}+{\sqrt {2}}{\left({\sqrt {2}}+{\sqrt {3}}\right)}+{\sqrt {5}}\\&=-5{\left({\sqrt {2}}^{3}+3{\sqrt {2}}^{2}{\sqrt {3}}+3{\sqrt {2}}{\sqrt {3}}^{2}+{\sqrt {3}}^{3}\right)}-{\left({\sqrt {2}}^{2}+2{\sqrt {2}}{\sqrt {3}}+{\sqrt {3}}^{2}\right)}+{\sqrt {2}}{\left({\sqrt {2}}+{\sqrt {3}}\right)}+{\sqrt {5}}\\&=-5{\left(2{\sqrt {2}}+6{\sqrt {3}}+9{\sqrt {2}}+3{\sqrt {3}}\right)}-{\left(2+2{\sqrt {2}}{\sqrt {3}}+3\right)}+2+{\sqrt {2}}{\sqrt {3}}+{\sqrt {5}}\\&=-5{\left(11{\sqrt {2}}+9{\sqrt {3}}\right)}-3-2{\sqrt {2}}{\sqrt {3}}+{\sqrt {2}}{\sqrt {3}}+{\sqrt {5}}\\&=-55{\sqrt {2}}-45{\sqrt {3}}-{\sqrt {2}}{\sqrt {3}}-3+{\sqrt {5}}.\,\end{aligned}}$

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