If quadratic equation `ax^(2) + bx + ab + bc + ca - a^(2) - b^(2) - c^(2) = 0` where `a, b, c` distinct reals, has imaginary roots than (A) `a+b+ab+bc+calta^(2)+b^(2)+c^(2)` (B)`a-b+ab+bc+cagta^(2)+b^(2)+c^(2)` (C)`4a+2b+ab+bc+calta^(2)+b^(2)+c^(2)` (D)`2(a+-3b)-9{(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}lt0`

Options

A. `a+b+ab+bc+calta^(2)+b^(2)+c^(2)`
B. `a-b+ab+bc+cagta^(2)+b^(2)+c^(2)`
C. `4a+2b+ab+bc+calta^(2)+b^(2)+c^(2)`
D. `2(a+-3b)-9{(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}lt0`

Explanation

To solve the quadratic equation \( ax^2 + bx + ab + bc + ca - a^2 - b^2 - c^2 = 0 \) and determine the conditions under which it has imaginary roots, we will follow these steps:

### Step 1: Identify the quadratic equation
The given quadratic equation is:
\[ ax^2 + bx + (ab + bc + ca - a^2 - b^2 - c^2) = 0 \]

### Step 2: Determine the discriminant
For a quadratic equation \( Ax^2 + Bx + C = 0 \), the roots are imaginary if the discriminant \( D = B^2 - 4AC < 0 \). Here, we identify:
- \( A = a \)
- \( B = b \)
- \( C = ab + bc + ca - a^2 - b^2 - c^2 \)

Thus, the discriminant \( D \) is:
\[ D = b^2 - 4a(ab + bc + ca - a^2 - b^2 - c^2) \]

### Step 3: Simplify the discriminant
We need to simplify the expression for \( D \):
\[ D = b^2 - 4a(ab + bc + ca - a^2 - b^2 - c^2) \]
Expanding this gives:
\[ D = b^2 - 4a(ab) - 4a(bc) - 4a(ca) + 4a(a^2) + 4a(b^2) + 4a(c^2) \]
\[ D = b^2 - 4a^2 + 4ab^2 + 4ac^2 - 4abc - 4abc \]

### Step 4: Set the discriminant less than zero
To find the conditions for imaginary roots, we set \( D < 0 \):
\[ b^2 - 4a(ab + bc + ca - a^2 - b^2 - c^2) < 0 \]

### Step 5: Analyze the options
Now we will analyze the provided options to see which one satisfies the condition \( D < 0 \).

1. **Option A**: \( a + b + ab + bc + ca < a^2 + b^2 + c^2 \)
2. **Option B**: \( a - b + ab + bc + ca < a^2 + b^2 + c^2 \)
3. **Option C**: \( 4a + 2b + ab + bc + ca < a^2 + b^2 + c^2 \)
4. **Option D**: \( 2(a - 3b) - 9((a-b)^2 + (b-c)^2 + (c-a)^2) < 0 \)

### Step 6: Determine which options are valid
- **Option A**: This condition can be derived from the discriminant analysis.
- **Option B**: This condition is not necessarily true as it can be greater than \( a^2 + b^2 + c^2 \).
- **Option C**: This condition can also be derived from the discriminant analysis.
- **Option D**: This condition can be valid under certain circumstances.

### Conclusion
The options that satisfy the condition for imaginary roots are:
- **Option A**
- **Option C**
- **Option D**