If the roots of the equation $(a - 1) (x^2 + x + 1)^2 = (a + 1) (x^4 + x^2 + 1)$ are real and distinct, then the value of $a$ $\in$
- $(-\infty, 3 ]$
- $(- \infty, - 2) \cup (2, \infty)$
- $[-2, 2]$
- $[-3, \infty )$
Dividing the equation by (x^2 + x + 1)^2 leads to (a - 1) = (a + 1) * ((x^4 + x^2 + 1) / (x^2 + x + 1)^2). Let t = (x^2 + 1) / x; the expression simplifies to a condition on the roots of a quadratic in t. Solving for real and distinct roots requires the discriminant to be positive, leading to the specified interval.
Expanding both sides and bringing all terms to one side gives the equation (a - 3)x^4 + (a - 1)x^3 + (a - 1)x + (a - 3) = 0. Dividing by (x^4 + 1) groups the terms into (a - 3) + (a - 1)(x^3 + x)/(x^4 + 1) = 0. Let k = (x^3 + x)/(x^4 + 1), where the maximum and minimum values of k are found by setting its derivative to zero to be 1/2 and -1/2. For the original equation to have real and distinct roots, the value a = (3 + k)/(1 + k) must yield at least four real roots, which happens when a is outside the interval of the critical values (-2, 2). Therefore, a must belong to the intervals (-infinity, -2) union (2, infinity).