Let $\alpha ,\beta$  be the roots of the quadratic equation ${\mathrm{x}}^2+\sqrt{6}\mathrm{x}+3=0.$ Then $\frac{{\mathrm{\alpha }}^{23}+{\mathrm{\beta }}^{23}+{\mathrm{\alpha }}^{14}+{\mathrm{\beta }}^{14}}{{\mathrm{\alpha }}^{15}+{\mathrm{\beta }}^{15}+{\mathrm{\alpha }}^{10}+{\mathrm{\beta }}^{10}}$ is equal to

$$\begin{gathered} {\mathrm{x}}^2+3=-\sqrt{6}\mathrm{x} \\ \Rightarrow {\mathrm{x}}^4+6{\mathrm{x}}^2+9=6{\mathrm{x}}^2\Rightarrow {\mathrm{x}}^4+9=0 \\ \frac{{\mathrm{\alpha }}^{23}+{\mathrm{\beta }}^{23}+{\mathrm{\alpha }}^{14}+{\mathrm{\beta }}^{14}}{{\mathrm{\alpha }}^{15}+{\mathrm{\beta }}^{15}+{\mathrm{\alpha }}^{10}+{\mathrm{\beta }}^{10}}=\frac{{\mathrm{\alpha }}^{14}({\mathrm{\alpha }}^9+1)+{\mathrm{\beta }}^{14}({\mathrm{\beta }}^9+1)}{{\mathrm{\alpha }}^{10}({\mathrm{\alpha }}^5+1)+{\mathrm{\beta }}^{10}({\mathrm{\beta }}^5+1)} \\ =\frac{(-729){\mathrm{\alpha }}^2(81\mathrm{\alpha }+1)-729{\mathrm{\beta }}^2(81\mathrm{\beta }+1)}{81{\mathrm{\alpha }}^2(-9\mathrm{\alpha }+1)+81{\mathrm{\beta }}^2(-9\mathrm{\beta }+1)} \\ =(-9)\left[\frac{81{\mathrm{\alpha }}^3+{\mathrm{\alpha }}^2+81{\mathrm{\beta }}^3+{\mathrm{\beta }}^2}{-9{\mathrm{\alpha }}^3+{\mathrm{\alpha }}^2-9{\mathrm{\beta }}^3+{\mathrm{\beta }}^2}\right] \\ \left(\mathrm{\alpha }+\mathrm{\beta }=-\sqrt{6};\mathrm{\alpha \beta }=3\Rightarrow {\mathrm{\alpha }}^2+{\mathrm{\beta }}^2=0\right) \\ =-9\frac{81({\mathrm{\alpha }}^3+{\mathrm{\beta }}^3)}{-9({\mathrm{\alpha }}^3+{\mathrm{\beta }}^3)}=81 \end{gathered}$$