To solve this equation, we need to find the values of x that satisfy the given equation.

First, we need to know that the cosine function has a period of 2π, which means that for any integer n, we have cos(x) = cos(x + 2nπ).

Given that cos(3π/4 + 2x) = -1, we can rewrite this as cos(3π/4 + 2x) = cos(π).

Since the cosine function is an even function, we can rewrite the equation as follows:

3π/4 + 2x = π + 2nπ (where n is an integer)

Now, we can solve for x:

3π/4 + 2x = π + 2nπ
2x = π + 2nπ - 3π/4
2x = 4π/4 + (8n - 3)/4π
2x = (4 + 8n - 3)/4π
2x = (8n + 1)/4
x = (8n + 1)/8

Therefore, the solutions for x are x = (8n + 1)/8, where n is an integer.