To solve the equation 2cos^2x - 5cos(-3/2π-x) + 1 = 0, we can start by simplifying the cosine terms.

Recall that cos(-θ) = cos(θ), so cos(-3/2π-x) = cos(3/2π + x).

Now the equation becomes:

2cos^2x - 5cos(3/2π + x) + 1 = 0

Next, we can expand the cosine term using the formula for cos(α + β):

cos(α + β) = cosα cosβ - sinα sinβ

cos(3/2π + x) = cos(3/2π) cos(x) - sin(3/2π) sin(x)
cos(3/2π) = 0
sin(3/2π) = -1

cos(3/2π + x) = 0 cos(x) - (-1) sin(x)
cos(3/2π + x) = sin(x)

Now the equation becomes:

2cos^2x - 5sinx + 1 = 0

This is a trigonometric equation that involves both cosine and sine terms. It's not possible to solve this equation algebraically. You may need to use numerical methods or a graphing calculator to find approximate solutions.