If `2x^(2) + y^(2) + 6x - 2xy + 9 = 0`, then the value of `(4x^(3) - y^(3) + x^(2)y^(2))` is :

Options

A. 0
B. 9
C. `-3`
D. `-9`

Answer

To solve the equation \(2x^2 + y^2 + 6x - 2xy + 9 = 0\) and find the value of \(4x^3 - y^3 + x^2y^2\), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ 2x^2 + y^2 + 6x - 2xy + 9 = 0 \]
We can rearrange it as:
\[ 2x^2 - 2xy + y^2 + 6x + 9 = 0 \]

### Step 2: Factor the quadratic expression
Notice that \(2x^2 - 2xy + y^2\) can be rewritten as:
\[ (x - y)^2 + 2x^2 + 6x + 9 = 0 \]
This can be factored further:
\[ (x - y)^2 + (x + 3)^2 = 0 \]

### Step 3: Solve the equation
For the sum of squares to be zero, both squares must equal zero:
1. \(x - y = 0 \Rightarrow x = y\)
2. \(x + 3 = 0 \Rightarrow x = -3\)

Since \(x = y\), we have:
\[ x = -3 \quad \text{and} \quad y = -3 \]

### Step 4: Substitute values into the expression
Now we substitute \(x = -3\) and \(y = -3\) into the expression \(4x^3 - y^3 + x^2y^2\):
\[ 4(-3)^3 - (-3)^3 + (-3)^2(-3)^2 \]

### Step 5: Calculate each term
Calculating each term:
- \(4(-3)^3 = 4 \times -27 = -108\)
- \(-(-3)^3 = 27\)
- \((-3)^2(-3)^2 = 9 \times 9 = 81\)

### Step 6: Combine the results
Now combine the results:
\[ -108 + 27 + 81 \]
Calculating this gives:
\[ -108 + 27 = -81 \]
\[ -81 + 81 = 0 \]

### Final Answer
Thus, the value of \(4x^3 - y^3 + x^2y^2\) is:
\[ \boxed{0} \]