To solve the equation \(\cos(x + 30^\circ) = 2 \cos x\), we use the cosine addition formula:

\(\cos(x + 30^\circ) = \cos x \cos 30^\circ - \sin x \sin 30^\circ.\)

Substitute \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\):

\(\frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x = 2 \cos x.\)

Rearrange to form:

\(\frac{\sqrt{3}}{2} \cos x - 2 \cos x = \frac{1}{2} \sin x.\)

Simplify:

\((\frac{\sqrt{3}}{2} - 2) \cos x = \frac{1}{2} \sin x.\)

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

\(\tan x = \frac{\frac{1}{2}}{2 - \frac{\sqrt{3}}{2}}.\)

Simplify the expression:

\(\tan x = \sqrt{3} - 4.\)

Find \(x\) using the arctangent function:

\(x = \tan^{-1}(\sqrt{3} - 4).\)

Calculate the solutions within the interval \(-180^\circ < x < 180^\circ\):

\(x \approx -66.2^\circ \text{ and } x \approx 113.8^\circ.\)