4cos²x - 8cosx - 5 = 0

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To solve this quadratic equation for cos(x), we can use the quadratic formula:

cos(x) = [-(-8) ± √((-8)² - 4(4)(-5))] / 2(4)

cos(x) = [8 ± √(64 + 80)] / 8

cos(x) = [8 ± √144] / 8

cos(x) = [8 ± 12] / 8

To find the two possible solutions for cos(x), we divide this into two cases:
1) cos(x) = (8 + 12) / 8 = 20/8 = 2.5
2) cos(x) = (8 - 12) / 8 = -4/8 = -0.5

However, the cosine function only ranges from -1 to 1, so the second solution (-0.5) is not valid.
Thus, the only solution for cos(x) is 2.5, but since this value is outside the valid range for cos(x), there are no real solutions to the original equation 4cos²x - 8cosx - 5 = 0.