If the mirror image of the point

Image of $(1,3,5)$ in the plane $4 x-5 y+2 z=8$ is $(\alpha, \beta, \gamma)$

$\Rightarrow \frac{\alpha-1}{4}=\frac{\beta-3}{-5}=\frac{\gamma-5}{2}=-2 \frac{(4(1)-5(3)+2(5)-8)}{4^{2}+5^{2}+2^{2}}=\frac{2}{5}$

$\therefore \alpha=1+4\left(\frac{2}{5}\right)=\frac{13}{5}$

$\beta=3-5\left(\frac{2}{5}\right)=1=\frac{5}{5}$

$\gamma=5+2\left(\frac{2}{5}\right)=\frac{29}{5}$

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