Consider the following statements in respect of the given equation.
`(x^(2) + 2)^(2) + 8x^(2) = 6x (x^(2) + 2)`
1. All the root of the equation are complex.
2. The sum of the all the roots of the equation is 6.
Which of the above statements is/are correct ?

Options

A. a) Only I
B. b) Only II
C. c) Both I and II
D. d) Neither I nor II

Answer

To solve the equation \((x^2 + 2)^2 + 8x^2 = 6x(x^2 + 2)\) and analyze the statements, we will follow these steps:

### Step 1: Expand the equation
We start by expanding both sides of the equation.

1. Left-hand side:
\[ (x^2 + 2)^2 + 8x^2 = (x^4 + 4x^2 + 4) + 8x^2 = x^4 + 12x^2 + 4 \]

2. Right-hand side:
\[ 6x(x^2 + 2) = 6x^3 + 12x \]

Now, we can rewrite the equation as:
\[ x^4 + 12x^2 + 4 = 6x^3 + 12x \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to set it to zero:
\[ x^4 - 6x^3 + 12x^2 - 12x + 4 = 0 \]

### Step 3: Analyze the roots of the polynomial
To determine the nature of the roots, we can use the discriminant or analyze the behavior of the polynomial.

1. **Finding the discriminant**: We can use the discriminant of the polynomial. However, for a quartic polynomial, this can be complex. Instead, we can check the values of the polynomial at certain points to see if it changes signs.

2. **Evaluate at specific points**:
- \(f(0) = 4\)
- \(f(1) = 1 - 6 + 12 - 12 + 4 = -1\)

Since \(f(0) > 0\) and \(f(1) < 0\), by the Intermediate Value Theorem, there is at least one real root between \(0\) and \(1\).

### Step 4: Sum of the roots
Using Vieta's formulas, the sum of the roots of the polynomial \(ax^4 + bx^3 + cx^2 + dx + e = 0\) is given by \(-\frac{b}{a}\).

Here, \(a = 1\) and \(b = -6\):
\[ \text{Sum of roots} = -\frac{-6}{1} = 6 \]

### Conclusion
Now we can evaluate the statements:
1. **Statement 1**: All the roots of the equation are complex. **False** (since we found at least one real root).
2. **Statement 2**: The sum of all the roots of the equation is 6. **True**.

Thus, the correct answer is that only Statement 2 is true.