Let α, β be the roots of the quadratic equation x ^ 2 + 6 x + 3 = 0...

To find the value of $\dfrac{{\alpha}^{23} + {\beta}^{23} + {\alpha}^{14} + {\beta}^{14}}{{\alpha}^{15} + {\beta}^{15} + {\alpha}^{10} + {\beta}^{10}}$,

Let ${a}_{n} = {\alpha}^{n} + {\beta}^{n}$

Hence,

$\dfrac{{\alpha}^{23} + {\beta}^{23} + {\alpha}^{14} + {\beta}^{14}}{{\alpha}^{15} + {\beta}^{15} + {\alpha}^{10} + {\beta}^{10}} = \dfrac{{a}_{23} + {a}_{14}}{{a}_{15} + {a}_{10}}$

Now, ${x}^{2} + \sqrt{6}x + 3 = 0$ has roots $\alpha \text{ and } \beta$

So, $x = \dfrac{- \sqrt{6} \pm \sqrt{- 6}}{2}$

$⇒ x = \sqrt{6}\left(\dfrac{- 1 \pm i}{2}\right)$

$⇒ x = \sqrt{3}\left(\dfrac{- 1 \pm i}{\sqrt{2}}\right)$

Hence, $\alpha = \sqrt{3}e\dfrac{i3\pi}{4}$ and $\beta = \sqrt{3}e\dfrac{i5\pi}{4}$

Now, solving $\dfrac{{\alpha}^{23} + {\beta}^{23} + {\alpha}^{14} + {\beta}^{14}}{{\alpha}^{15} + {\beta}^{15} + {\alpha}^{10} + {\beta}^{10}}$

$= \left[\dfrac{{\left(\sqrt{3}\right)}^{23}\left({e}^{i23 \times \dfrac{3\pi}{4}} + {e}^{i23 \times \dfrac{5\pi}{4}}\right) + {\left(\sqrt{3}\right)}^{14}\left({e}^{i14 \times \dfrac{3\pi}{4}} + {e}^{i14 \times \dfrac{5\pi}{4}}\right)}{{\left(\sqrt{3}\right)}^{15}\left({e}^{i15 \times \dfrac{3\pi}{4}} + {e}^{i15 \times \dfrac{5\pi}{4}}\right) + {\left(\sqrt{3}\right)}^{10}\left({e}^{i10 \times \dfrac{3\pi}{4}} + {e}^{i \times 10 \times \dfrac{5\pi}{4}}\right)}\right]$

$= {\left(\sqrt{3}\right)}^{4}\left[\dfrac{{\left(\sqrt{3}\right)}^{9}\left({e}^{i23 \times \dfrac{3\pi}{4}} + {e}^{i23 \times \dfrac{5\pi}{4}}\right) + \left({e}^{i14 \times \dfrac{3\pi}{4}} + {e}^{i14 \times \dfrac{5\pi}{4}}\right)}{{\left(\sqrt{3}\right)}^{5}\left({e}^{i15 \times \dfrac{3\pi}{4}} + {e}^{i15 \times \dfrac{5\pi}{4}}\right) + \left({e}^{i10 \times \dfrac{3\pi}{4}} + {e}^{i \times 10 \times \dfrac{5\pi}{4}}\right)}\right]$

$= 9 \times \left(\dfrac{{\left(\sqrt{3}\right)}^{9}\left(\dfrac{1 + i - 1 + i}{\sqrt{2}}\right) + 0}{{\left(\sqrt{3}\right)}^{5}\left(\dfrac{1 + i - 1 + i}{\sqrt{2}}\right) + 0}\right)$\left\{\text{as}{e}^{i\dfrac{21\pi}{2}} + {e}^{i\dfrac{35\pi}{2}} = 0 + isin\dfrac{21\pi}{2} + 0 + isin\dfrac{35\pi}{2} = i - i = 0\right\}$</p><p>$= 9 \times \left(\dfrac{{\left(\sqrt{3}\right)}^{9}\left(\dfrac{2i}{\sqrt{2}}\right)}{{\left(\sqrt{3}\right)}^{5}\left(\dfrac{2i}{\sqrt{2}}\right)}\right)$</p><p>$= 9 \times {\left(\sqrt{3}\right)}^{4}$</p><p>$= 81