First, calculate fg(x):

fg(x) = f(g(x)) = -3(-x² - 1)² + 2 = -3(x⁴ + 2x² + 1) + 2 = -3x⁴ - 6x² - 3 + 2 = -3x⁴ - 6x² - 1.

Next, calculate gf(x):

gf(x) = g(f(x)) = -(-3x² + 2)² - 1 = -(9x⁴ - 12x² + 4) - 1 = -9x⁴ + 12x² - 5.

Substitute into the equation:

fg(x) - gf(x) + 8 = (-3x⁴ - 6x² - 1) - (-9x⁴ + 12x² - 5) + 8 = 6x⁴ - 18x² + 12 = 0.

Factor the equation:

\(6(x⁴ - 3x² + 2) = 0.\)

\(Let y = x², then y² - 3y + 2 = 0.\)

\(Factor the quadratic: (y - 1)(y - 2) = 0.\)

\(So, y = 1 or y = 2.\)

\(Thus, x² = 1 or x² = 2.\)

\(Therefore, x = -1 or x = -\sqrt{2}.\)