If α, β are the roots of the equation x ^ 2 - (5 + 3 ^ _ 3 5 - 5 ^...

Given $\alpha, \beta$ are the roots of the equation ${x}^{2} - \left(5 + {3}^{\sqrt{\log_{3}5}} - {5}^{\sqrt{\log_{5}3}}\right)x + 3\left({3}^{\left(\log_{3}5\right)\dfrac{1}{3}} - {5}^{\left(\log_{5}3\right)\dfrac{2}{3}} - 1\right) = 0$

Now ${3}^{\sqrt{\log_{3}5}} - {5}^{\sqrt{\log_{5}3}} = {3}^{\sqrt{\log_{3}5}} - {\left({3}^{\log_{3}5}\right)}^{\sqrt{\log_{5}3}}$

$= {3}^{\sqrt{\log_{3}5}} - {3}^{\sqrt{\log_{3}5}} = 0$

Also ${3}^{\left(\log_{3}5\right)\dfrac{1}{3}} - {5}^{\left(\log_{5}3\right)\dfrac{2}{3}} = {5}^{\left(\log_{5}3\right)\dfrac{2}{3}} - {5}^{\left(\log_{5}3\right)\dfrac{2}{3}}$

$= 0$

So, given equation becomes ${x}^{2} - 5x - 3 = 0$

i.e. $\alpha + \beta = 5; \alpha\beta = - 3$

Now if the roots are $\alpha + \dfrac{1}{\beta}$ and $\beta + \dfrac{1}{\alpha}$

i.e., $\dfrac{\alpha\beta + 1}{\beta} \text{ and } \dfrac{\alpha\beta + 1}{\alpha}$

i.e., $\dfrac{- 2}{\beta} \text{ and } \dfrac{- 2}{\alpha}$

Then let $\dfrac{- 2}{\alpha} = t ⇒ \alpha = \dfrac{- 2}{t}$

As ${\alpha}^{2} - 5\alpha - 3 = 0$

$⇒ {\left(\dfrac{- 2}{t}\right)}^{2} - 5\left(\dfrac{- 2}{t}\right) - 3 = 0$

$⇒ \dfrac{4}{{t}^{2}} + \dfrac{10}{t} - 3 = 0$

$⇒ 3{t}^{2} - 10t - 4 = 0$

i.e., $3{x}^{2} - 10x - 4 = 0$ is the required equation.