Official solution
Image of (1, 3, 5) in the plane 4x - 5y + 2z = 8 is (\[\alpha\], \[\beta\], \[\gamma\])
\[ \Rightarrow {{\alpha - 1} \over 4} = {{\beta - 3} \over { - 5}} = {{\gamma - 5} \over 2} = - {{(4(1) - 5(3) + 2(5) - 8)} \over {{4^2} + {5^2} + {2^2}}} = {2 \over 5}\]
\[ \therefore \] \[\alpha = 1 + 4\left( {{2 \over 5}} \right) = {{13} \over 5}\]
\[\beta = 3 - 5\left( {{2 \over 5}} \right) = 1 = {5 \over 5}\]
\[\gamma = 5 + 2\left( {{2 \over 5}} \right) = {{29} \over 5}\]
Thus, \[5(\alpha + \beta + \gamma ) = 5\left( {{{13} \over 5} + {5 \over 5} + {{29} \over 5}} \right) = 47\]