The range of values of a for which all the roots of the equation $\left(a-1\right)\left(1+x+x^2\right)^2=\left(a+1\right)\left(1+x^2+x^4\right)$ imaginary is

For the roots to be imaginary, the discriminant of the quadratic in terms of (x + 1/x) must be negative. The expression simplifies to (a-1)(x + 1/x)^2 - (a+1)(x + 1/x + (x^2 + 1/x^2)) which simplifies to a quadratic in y = x + 1/x where the discriminant condition requires a < -2 or a > 2. Since the range of y = x + 1/x is (-infinity, -2] union [2, infinity), requiring the discriminant to be negative for all y in this range forces the condition that a <= -2. Therefore, the range of values of a is (-infinity, -2].

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