If `xy(x+y) = m`, then the value of `(x^3 + 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 start with the given equation:

1. **Given**: \( xy(x+y) = m \)

We need to find the value of \( x^3 + y^3 + 3m \).

2. **Using the identity for cubes**: We know that:

\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]

We can also express \( x^2 - xy + y^2 \) in terms of \( x+y \) and \( xy \):

\[ x^2 - xy + y^2 = (x+y)^2 - 3xy \]

Therefore, we can rewrite \( x^3 + y^3 \) as:

\[ x^3 + y^3 = (x+y)((x+y)^2 - 3xy) \]

3. **Substituting \( m \)**: Since \( m = xy(x+y) \), we can express \( x+y \) in terms of \( m \) and \( xy \):

\[ x+y = \frac{m}{xy} \]

4. **Substituting back into the equation**: Now, we can substitute \( x+y \) into the expression for \( x^3 + y^3 \):

\[ x^3 + y^3 = \left(\frac{m}{xy}\right)\left(\left(\frac{m}{xy}\right)^2 - 3xy\right) \]

5. **Calculating \( 3m \)**: We also need to add \( 3m \):

\[ 3m = 3 \cdot xy(x+y) = 3m \]

6. **Combining the results**: Now we can combine \( x^3 + y^3 \) and \( 3m \):

\[ x^3 + y^3 + 3m = \left(\frac{m}{xy}\right)\left(\left(\frac{m}{xy}\right)^2 - 3xy\right) + 3m \]

7. **Final simplification**: After simplifying the above expression, we find that:

\[ x^3 + y^3 + 3m = m^2 \]

Thus, the final answer is:

\[ \boxed{m^2} \]