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. less than `(-4ba)`
B. greater than `4ab`
C. less than `4ab`
D. greater than `(-4ab)`

Explanation

To solve the problem step-by-step, we need to analyze the given quadratic equation and the expression we need to evaluate.

### Step 1: Understand the condition for imaginary roots
The roots of the quadratic equation \( b x^2 + c x + a = 0 \) are imaginary if the discriminant \( D \) is less than zero. The discriminant is given by:
\[ D = c^2 - 4ab \]
For the roots to be imaginary, we have:
\[ c^2 - 4ab < 0 \implies c^2 < 4ab \]

**Hint:** Check the condition for imaginary roots using the discriminant.

### Step 2: Analyze the expression
We need to analyze the expression:
\[ 3b^2x^2 + 6bcx + 2c^2 \]
This is a quadratic expression in terms of \( x \).

### Step 3: Identify the coefficients
The coefficients of the quadratic expression are:
- \( A = 3b^2 \)
- \( B = 6bc \)
- \( C = 2c^2 \)

### Step 4: Find the minimum value of the quadratic expression
The minimum value of a quadratic expression \( Ax^2 + Bx + C \) occurs at:
\[ x = -\frac{B}{2A} = -\frac{6bc}{2 \cdot 3b^2} = -\frac{bc}{b^2} = -\frac{c}{b} \quad (\text{assuming } b \neq 0) \]

### Step 5: Substitute \( x \) back into the expression
Now, we substitute \( x = -\frac{c}{b} \) back into the expression to find its minimum value:
\[ 3b^2\left(-\frac{c}{b}\right)^2 + 6bc\left(-\frac{c}{b}\right) + 2c^2 \]
Calculating each term:
1. \( 3b^2\left(\frac{c^2}{b^2}\right) = 3c^2 \)
2. \( 6bc\left(-\frac{c}{b}\right) = -6c^2 \)
3. \( 2c^2 = 2c^2 \)

Combining these:
\[ 3c^2 - 6c^2 + 2c^2 = -c^2 \]

### Step 6: Relate the minimum value to \( -4ab \)
From the condition we derived earlier, we know:
\[ c^2 < 4ab \implies -c^2 > -4ab \]
Thus, the minimum value of the expression \( 3b^2x^2 + 6bcx + 2c^2 \) is:
\[ -c^2 > -4ab \]

### Conclusion
Therefore, we conclude that:
\[ 3b^2x^2 + 6bcx + 2c^2 > -4ab \]

The correct answer is:
**(1) greater than -4ab**