Start with the given equation:

\(6 \sin^2 x - 5 \cos^2 x = 2 \sin^2 x + \cos^2 x\)

Rearrange terms:

\(6 \sin^2 x - 2 \sin^2 x = 5 \cos^2 x + \cos^2 x\)

\(4 \sin^2 x = 6 \cos^2 x\)

Divide both sides by \(\cos^2 x\):

\(\frac{4 \sin^2 x}{\cos^2 x} = 6\)

\(4 \tan^2 x = 6\)

\(\tan^2 x = \frac{6}{4} = 1.5\)

\(\tan x = \pm \sqrt{1.5}\)

\(\tan x = \pm 1.225\)

Find the angles \(x\) in the range \(0^\circ \leq x \leq 360^\circ\):

\(x = 50.8^\circ, 129.2^\circ, 230.8^\circ, 309.2^\circ\)