(B) (2, 9/4) If two roots of the equation `(a-1) (x^2 + x + 1 )^2-(a + 1) (x^4 + x^2 + 1) = 0` are real and distinct, then a lies in the interval

Options

A. `(- oo, 3]`
B. `(-oo, -2) uu (2, oo)`
C. `[-2, 2]`
D. `[-3, oo)`

Explanation

To solve the equation \((a-1)(x^2 + x + 1)^2 - (a + 1)(x^4 + x^2 + 1) = 0\) and determine the values of \(a\) for which two roots are real and distinct, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ (a-1)(x^2 + x + 1)^2 - (a + 1)(x^4 + x^2 + 1) = 0 \]

### Step 2: Expand the terms
We need to expand both sides of the equation. The term \((x^2 + x + 1)^2\) can be expanded as:
\[ (x^2 + x + 1)(x^2 + x + 1) = x^4 + 2x^3 + 3x^2 + 2x + 1 \]
Then, substituting this back into the equation gives:
\[ (a-1)(x^4 + 2x^3 + 3x^2 + 2x + 1) - (a + 1)(x^4 + x^2 + 1) = 0 \]

### Step 3: Combine like terms
Now we will distribute \(a-1\) and \(a+1\) across their respective polynomials:
\[ (a-1)x^4 + 2(a-1)x^3 + 3(a-1)x^2 + 2(a-1)x + (a-1) - (a+1)x^4 - (a+1)x^2 - (a+1) = 0 \]
Combining like terms yields:
\[ [(a-1) - (a+1)]x^4 + [2(a-1)]x^3 + [(3(a-1) - (a+1))]x^2 + [2(a-1)]x + [(a-1) - (a+1)] = 0 \]
This simplifies to:
\[ -2x^4 + 2(a-1)x^3 + (2a - 4)x^2 + 2(a-1)x - 2 = 0 \]

### Step 4: Factor out common terms
Factoring out \(-2\) gives:
\[ x^4 - (a-1)x^3 - (2a - 4)x^2 - (a-1)x + 1 = 0 \]

### Step 5: Analyze the roots
For the roots to be real and distinct, we need to analyze the discriminant of the resulting polynomial. The discriminant \(D\) must be greater than 0:
\[ D > 0 \]

### Step 6: Set up the discriminant condition
The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:
\[ D = B^2 - 4AC \]
For our polynomial, we need to ensure that the discriminant of the quadratic part is greater than zero.

### Step 7: Solve the inequality
Setting up the inequality from the discriminant:
\[ (a-1)^2 - 4(1)(1) > 0 \]
This simplifies to:
\[ (a-1)^2 > 4 \]
Taking square roots gives:
\[ |a-1| > 2 \]
This leads to two cases:
1. \(a - 1 > 2 \Rightarrow a > 3\)
2. \(a - 1 < -2 \Rightarrow a < -1\)

### Step 8: Combine intervals
Thus, the values of \(a\) that satisfy the condition for real and distinct roots are:
\[ a < -1 \quad \text{or} \quad a > 3 \]

### Conclusion
The intervals for \(a\) are:
\[ (-\infty, -1) \cup (3, \infty) \]

### Final Answer
The values of \(a\) for which two roots of the equation are real and distinct lie in the intervals:
\[ (-\infty, -1) \cup (3, \infty) \]