If `xy (x + y) = m ,` then the value of `x ^(2) + y ^(3) + 3m` is ?

Options

A. `( m ^(3))/( xy )`
B. `(m ^(3))/(( x + y ) ^(3))`
C. `(m ^(3))/( x ^(3) y ^(3))`
D. `mx ^(3) y ^(3)`

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 + 3m \) given that \( xy(x + y) = m \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ xy(x + y) = m \]
This is our equation (1).

2. **Express \( m \) in terms of \( x \) and \( y \)**:
From equation (1), we have:
\[ m = xy(x + y) \]

3. **Find \( x^2 + y^2 \)**:
We know the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Let \( s = x + y \) and \( p = xy \). Then:
\[ x^2 + y^2 = s^2 - 2p \]

4. **Substituting \( m \)**:
Since \( m = p \cdot s \), we can express \( p \) in terms of \( m \) and \( s \):
\[ p = \frac{m}{s} \]

5. **Substituting \( p \) into the equation for \( x^2 + y^2 \)**:
Now substitute \( p \) into the equation for \( x^2 + y^2 \):
\[ x^2 + y^2 = s^2 - 2\left(\frac{m}{s}\right) \]

6. **Now find \( x^2 + y^2 + 3m \)**:
We need to find:
\[ x^2 + y^2 + 3m = \left(s^2 - 2\left(\frac{m}{s}\right)\right) + 3m \]
Simplifying this:
\[ = s^2 - \frac{2m}{s} + 3m \]

7. **Combine the terms**:
To combine the terms, we can write:
\[ = s^2 + 3m - \frac{2m}{s} \]

8. **Final expression**:
This is the expression for \( x^2 + y^2 + 3m \). We can leave it in this form or evaluate further if specific values for \( x \) and \( y \) are provided.

### Final Answer:
The value of \( x^2 + y^2 + 3m \) is:
\[ s^2 + 3m - \frac{2m}{s} \]