If the roots of the equation `b x^2+""c x""+""a""=""0` be imaginary, then for all real values of x, the expression `3b^2x^2+""6b c x""+""2c^2` is (1) greater than 4ab (2) less than 4ab (3) greater than `4a b` (4) less than `4a b`

Options

A. greater than 4ab.
B. less then 4ab
C. grreater than - 4ab.
D. less than - 4ab.

Explanation

To solve the problem, we need to analyze the given quadratic equation and the expression provided.

### Step-by-Step Solution:

1. **Identify the Condition for Imaginary Roots**:
The roots of the quadratic equation \( bx^2 + cx + a = 0 \) are imaginary if the discriminant is less than zero. The discriminant \( D \) is given by:
\[ D = c^2 - 4ab \]
For the roots to be imaginary:
\[ c^2 - 4ab < 0 \quad \Rightarrow \quad c^2 < 4ab \]

**Hint**: Remember that the discriminant must be negative for the roots to be imaginary.

2. **Consider the Expression**:
We need to analyze the expression:
\[ E = 3b^2x^2 + 6bcx + 2c^2 \]
This is a quadratic expression in terms of \( x \).

3. **Find the Discriminant of the Expression**:
The discriminant of the expression \( E \) can be computed as follows:
\[ D_E = (6bc)^2 - 4(3b^2)(2c^2) \]
Simplifying this:
\[ D_E = 36b^2c^2 - 24b^2c^2 = 12b^2c^2 \]

**Hint**: The discriminant of a quadratic expression helps determine the nature of its roots.

4. **Determine the Nature of the Expression**:
Since \( D_E = 12b^2c^2 \) is always non-negative (as \( b^2 \) and \( c^2 \) are always non-negative), the expression \( E \) will always have real roots.

5. **Find the Minimum Value of the Expression**:
The minimum value of a quadratic expression \( ax^2 + bx + c \) occurs at:
\[ x = -\frac{b}{2a} \]
For our expression:
\[ x = -\frac{6bc}{2 \cdot 3b^2} = -\frac{c}{b} \]
Plugging this value back into \( E \):
\[ E\left(-\frac{c}{b}\right) = 3b^2\left(-\frac{c}{b}\right)^2 + 6bc\left(-\frac{c}{b}\right) + 2c^2 \]
Simplifying:
\[ = 3b^2\frac{c^2}{b^2} - 6c^2 + 2c^2 = 3c^2 - 6c^2 + 2c^2 = -c^2 \]

6. **Compare with \( 4ab \)**:
Since we know \( c^2 < 4ab \), we can conclude:
\[ -c^2 > -4ab \quad \Rightarrow \quad E \text{ is greater than } 4ab \]

### Conclusion:
Thus, the expression \( 3b^2x^2 + 6bcx + 2c^2 \) is greater than \( 4ab \).

The correct option is:
(1) greater than \( 4ab \).