If $$\alpha, \beta$$ are the roots of the equation $$x^2 - (5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}})x + 3(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1) = 0$$ then the equation, whose roots are $$\alpha + \dfrac{1}{\beta}$$ and $$\beta + \dfrac{1}{\alpha}$$, is

We first simplify the coefficients in the given equation $$x^2 - \left(5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}}\right)x + 3\left(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1\right) = 0$$.

Let $$t = \log_3 5$$, so $$\log_5 3 = \dfrac{1}{t}$$ and $$5 = 3^t$$.

For the coefficient of $$x$$: $$3^{\sqrt{t}}$$ and $$5^{\sqrt{1/t}} = (3^t)^{1/\sqrt{t}} = 3^{t/\sqrt{t}} = 3^{\sqrt{t}}$$. These are equal, so $$3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}} = 0$$. The coefficient of $$x$$ is $$-(5 + 0) = -5$$.

For the constant term: $$3^{t^{1/3}}$$ and $$5^{(1/t)^{2/3}} = (3^t)^{t^{-2/3}} = 3^{t \cdot t^{-2/3}} = 3^{t^{1/3}}$$. These are also equal, so $$3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} = 0$$. The constant term is $$3(0 - 1) = -3$$.

The simplified equation is $$x^2 - 5x - 3 = 0$$, with roots $$\alpha + \beta = 5$$ and $$\alpha\beta = -3$$.

The new roots are $$\alpha + \dfrac{1}{\beta}$$ and $$\beta + \dfrac{1}{\alpha}$$.

Sum of new roots $$= (\alpha + \beta) + \left(\dfrac{1}{\alpha} + \dfrac{1}{\beta}\right) = (\alpha + \beta) + \dfrac{\alpha + \beta}{\alpha\beta} = 5 + \dfrac{5}{-3} = 5 - \dfrac{5}{3} = \dfrac{10}{3}$$

Product of new roots $$= \left(\alpha + \dfrac{1}{\beta}\right)\left(\beta + \dfrac{1}{\alpha}\right) = \alpha\beta + 1 + 1 + \dfrac{1}{\alpha\beta} = -3 + 2 + \dfrac{1}{-3} = -1 - \dfrac{1}{3} = -\dfrac{4}{3}$$

The equation with roots having sum $$\dfrac{10}{3}$$ and product $$-\dfrac{4}{3}$$ is:

$$x^2 - \dfrac{10}{3}x - \dfrac{4}{3} = 0$$

Multiplying by 3: $$3x^2 - 10x - 4 = 0$$.

The correct answer is Option B: $$3x^2 - 10x - 4 = 0$$.