If `m + n = - 2 ` then the value of ` m^(3) + n^(3) - 6` mn is

Options

A. 8
B. 4
C. `-8`
D. `-4`

Answer

To solve the problem, we need to find the value of \( m^3 + n^3 - 6mn \) given that \( m + n = -2 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
We know that:
\[ m^3 + n^3 = (m+n)(m^2 - mn + n^2) \]
We can also express \( m^2 + n^2 \) in terms of \( m+n \) and \( mn \):
\[ m^2 + n^2 = (m+n)^2 - 2mn \]

2. **Substitute \( m+n \)**:
Since \( m+n = -2 \), we can substitute this into the identity:
\[ m^3 + n^3 = (-2)(m^2 - mn + n^2) \]

3. **Calculate \( m^2 + n^2 \)**:
Substitute \( m+n \) into the equation for \( m^2 + n^2 \):
\[ m^2 + n^2 = (-2)^2 - 2mn = 4 - 2mn \]

4. **Substitute \( m^2 + n^2 \) back into the sum of cubes**:
Now, we can substitute \( m^2 + n^2 \) into our equation for \( m^3 + n^3 \):
\[ m^3 + n^3 = (-2)((4 - 2mn) - mn) = (-2)(4 - 3mn) \]
This simplifies to:
\[ m^3 + n^3 = -8 + 6mn \]

5. **Combine with the original expression**:
Now we need to find \( m^3 + n^3 - 6mn \):
\[ m^3 + n^3 - 6mn = (-8 + 6mn) - 6mn \]
This simplifies to:
\[ m^3 + n^3 - 6mn = -8 \]

### Final Answer:
Thus, the value of \( m^3 + n^3 - 6mn \) is \( -8 \).

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