Suppose `x, y in R`. If `x^(2) + y + 4i` is conjugate of `-3 + x^(2) yi`, then maximum possible value of `(|x| + |y|)^(2)` is equal to _____________.

Answer

To solve the problem, we need to analyze the given conditions involving complex numbers. We start with the expression given in the problem and use the properties of conjugates.

### Step-by-Step Solution:

1. **Understanding the Conjugate**:
We have the expression \( x^2 + y + 4i \) which is said to be the conjugate of \( -3 + x^2 yi \). The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( -3 + x^2 yi \) is \( -3 - x^2 yi \).

2. **Setting Up the Equation**:
Since \( x^2 + y + 4i \) is the conjugate of \( -3 + x^2 yi \), we can equate the real and imaginary parts:
- Real part: \( x^2 + y = -3 \) (Equation 1)
- Imaginary part: \( 4 = -x^2 y \) (Equation 2)

3. **Solving the Equations**:
From Equation 1, we can express \( y \) in terms of \( x^2 \):
\[ y = -3 - x^2 \]
Now, substitute this expression for \( y \) into Equation 2:
\[ 4 = -x^2(-3 - x^2) \]
Simplifying this gives:
\[ 4 = 3x^2 + x^4 \]
Rearranging leads to:
\[ x^4 + 3x^2 - 4 = 0 \]

4. **Letting \( z = x^2 \)**:
We can let \( z = x^2 \), which transforms our equation into a quadratic:
\[ z^2 + 3z - 4 = 0 \]
We can solve this using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm 5}{2} \]
This gives us two possible values for \( z \):
\[ z = 1 \quad \text{or} \quad z = -4 \]
Since \( z = x^2 \) must be non-negative, we take \( z = 1 \) which implies \( x^2 = 1 \) and hence \( x = \pm 1 \).

5. **Finding \( y \)**:
Substituting \( x^2 = 1 \) back into Equation 1:
\[ y = -3 - 1 = -4 \]

6. **Calculating \( |x| + |y| \)**:
Now we calculate \( |x| + |y| \):
\[ |x| = 1 \quad \text{and} \quad |y| = 4 \]
Therefore,
\[ |x| + |y| = 1 + 4 = 5 \]

7. **Finding the Maximum Value**:
We need to find \( (|x| + |y|)^2 \):
\[ (|x| + |y|)^2 = 5^2 = 25 \]

Thus, the maximum possible value of \( (|x| + |y|)^2 \) is \( \boxed{25} \).