Let `a, b, c in R` with `a gt 0` such that the equation `ax^(2) + bcx + b^(3) + c^(3) - 4abc = 0` has non-real roots.
If `P(x) = ax^(2) + bx + c and Q(x) = ax^(2) + cx + b`, then (a) `P(x) gt 0` for all `x in R` and `Q(x) lt 0` for all `x in R`. (b) `P(x) lt 0` for all `x in R and Q(x) gt 0` for all `x in R`. (c) neither `P(x) gt 0` for all `x in R` nor `Q(x) gt 0` for all `x in R`. (d) exactly one of P(x) or Q(x) is positive for all real x.

Options

A. `P(x) gt 0` for all `x in R` and `Q(x) lt 0` for all `x in R`.
B. `P(x) lt 0` for all `x in R and Q(x) gt 0` for all `x in R`.
C. neither `P(x) gt 0` for all `x in R` nor `Q(x) gt 0` for all `x in R`.
D. exactly one of P(x) or Q(x) is positive for all real x.

Explanation

To solve the problem, we need to analyze the given equation and the polynomials \( P(x) \) and \( Q(x) \) based on the conditions provided.

### Step 1: Analyze the given equation
The equation given is:

\[ a x^2 + b c x + (b^3 + c^3 - 4abc) = 0 \]

We know that for this quadratic equation to have non-real roots, its discriminant must be less than zero. The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:

\[ D = B^2 - 4AC \]

For our equation, we identify:

- \( A = a \)
- \( B = bc \)
- \( C = b^3 + c^3 - 4abc \)

Thus, the discriminant becomes:

\[ D = (bc)^2 - 4a(b^3 + c^3 - 4abc) \]

### Step 2: Set up the condition for non-real roots
We need this discriminant to be less than zero:

\[ (bc)^2 - 4a(b^3 + c^3 - 4abc) < 0 \]

### Step 3: Simplify the discriminant condition
Expanding the discriminant gives:

\[ b^2c^2 - 4ab^3 - 4ac^3 + 16a^2bc < 0 \]

Rearranging this, we can factor it as:

\[ (b^2c^2 - 4ab^3) - 4a(c^3 - 4abc) < 0 \]

### Step 4: Analyze \( P(x) \) and \( Q(x) \)
Now, we define the polynomials:

\[ P(x) = ax^2 + bx + c \]
\[ Q(x) = ax^2 + cx + b \]

Next, we need to find the discriminants of \( P(x) \) and \( Q(x) \):

1. **Discriminant of \( P(x) \)**:
\[ D_P = b^2 - 4ac \]

2. **Discriminant of \( Q(x) \)**:
\[ D_Q = c^2 - 4ab \]

### Step 5: Analyze the relationship between \( D_P \) and \( D_Q \)
From our earlier analysis, we know:

\[ (bc)^2 - 4a(b^3 + c^3 - 4abc) < 0 \]

This implies that the product of the discriminants \( D_P \) and \( D_Q \) must have opposite signs. Therefore, we can conclude that:

- Either \( D_P > 0 \) and \( D_Q < 0 \)
- Or \( D_P < 0 \) and \( D_Q > 0 \)

This means that exactly one of the polynomials \( P(x) \) or \( Q(x) \) has real roots.

### Conclusion
Thus, the correct answer is:

**(d)** Exactly one of \( P(x) \) or \( Q(x) \) is positive for all real \( x \).