If the mirror image of the point $$(1, 3, 5)$$ with respect to the plane $$4x - 5y + 2z = 8$$ is $$(\alpha, \beta, \gamma)$$, then $$5(\alpha + \beta + \gamma)$$ equals:

The mirror image of a point $$(x_0, y_0, z_0)$$ with respect to the plane $$ax + by + cz = d$$ is given by $$(x_0, y_0, z_0) - \frac{2(ax_0 + by_0 + cz_0 - d)}{a^2 + b^2 + c^2}(a, b, c)$$.

Here $$(x_0, y_0, z_0) = (1, 3, 5)$$ and the plane is $$4x - 5y + 2z = 8$$. We compute $$4(1) - 5(3) + 2(5) - 8 = 4 - 15 + 10 - 8 = -9$$ and $$a^2 + b^2 + c^2 = 16 + 25 + 4 = 45$$.

The scaling factor is $$\frac{2(-9)}{45} = \frac{-18}{45} = -\frac{2}{5}$$.

So $$(\alpha, \beta, \gamma) = (1, 3, 5) - \left(-\frac{2}{5}\right)(4, -5, 2) = (1, 3, 5) + \frac{2}{5}(4, -5, 2) = \left(1 + \frac{8}{5},\ 3 - \frac{10}{5},\ 5 + \frac{4}{5}\right) = \left(\frac{13}{5},\ 1,\ \frac{29}{5}\right)$$.

Therefore, $$5(\alpha + \beta + \gamma) = 5\left(\frac{13}{5} + 1 + \frac{29}{5}\right) = 13 + 5 + 29 = 47$$.