To solve the equation \(4 \cos(2\theta + \pi) + 8 = 12\), we begin by isolating the trigonometric function. First, we subtract 8 from both sides:

\(4 \cos(2\theta + \pi) = 12 - 8\)

This simplifies to:

\(4 \cos(2\theta + \pi) = 4\)

Next, we divide both sides by 4 to isolate the cosine function:

\(\cos(2\theta + \pi) = 1\)

Now, we need to determine the angles for which the cosine equals 1. Referring to the unit circle, we find that the cosine function equals 1 at the angle \(0\). Therefore, we set:

\(2\theta + \pi = 0\)

Since we are looking for all solutions and not just a single angle, we add \(2\pi n\) (where \(n\) is any integer) to account for the periodic nature of the cosine function:

\(2\theta + \pi = 2\pi n\)

Next, we isolate \(2\theta\) by subtracting \(\pi\) from both sides:

\(2\theta = 2\pi n - \pi\)

To solve for \(\theta\), we divide both sides by 2:

\(\theta = \pi n - \frac{\pi}{2}\)

This final expression, \(\theta = \pi n - \frac{\pi}{2}\), represents all solutions to the original equation. By following these systematic steps, we can confidently arrive at the correct answer.