Step-by-step Solution:

\[ \begin{aligned} & 4 \cos^2 x + 6 \sin^2 x = 5 \\ & 4 \cos^2 x + 4 \sin^2 x + 2 \sin^2 x = 5 \\ & 4 + 2 \sin^2 x = 5 \quad \text{or} \quad 2 \sin^2 x = 1. \end{aligned} \] \[ \begin{aligned} & \sin^2 x = \frac{1}{2} \quad \text{or} \quad \sin^2 x = \left(\frac{1}{\sqrt{2}}\right)^2, \\ & \sin^2 x = \sin^2 \frac{\pi}{4} \Rightarrow x = n\pi \pm \frac{\pi}{4}. \end{aligned} \]