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Actually, Sage can solve your form symbolically as soon as you replace 6.0 with 6 and apply simplify_rectform:
sage: ((x^2 - 6)*x^2 - 4*x*(e^(2/5*I*pi) + e^(-2/5*I*pi)) - e^(4/5*I*pi) - e^(-4/5*I*pi) - 1).simplify_rectform()
(x^2 - 6)*x^2 - 2*x*(sqrt(5) - 1) + 1/2*sqrt(5) - 1/2
sage: solve(_,x)
[x == -1/2*sqrt(-2*sqrt(5) + 10) - 1, x == 1/2*sqrt(-2*sqrt(5) + 10) - 1, x == -1/2*sqrt(2*sqrt(5) + 6) + 1, x == 1/2*sqrt(2*sqrt(5) + 6) + 1]
