If ` m=- 4 , n=-2` then the value of `m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3` is

Options

A. `-126`
B. `124`
C. `-124`
D. `126`

Answer

To find the value of the expression \( m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3 \) given \( m = -4 \) and \( n = -2 \), we can substitute the values of \( m \) and \( n \) into the expression and simplify step by step.

### Step-by-Step Solution:

1. **Substitute the values of m and n into the expression:**
\[ m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3 = (-4)^3 - 3(-4)^2 + 3(-4) + 3(-2) + 3(-2)^2 + (-2)^3 \]

2. **Calculate each term:**
- Calculate \( (-4)^3 \):
\[ (-4)^3 = -64 \]
- Calculate \( -3(-4)^2 \):
\[ -3(-4)^2 = -3 \times 16 = -48 \]
- Calculate \( 3(-4) \):
\[ 3(-4) = -12 \]
- Calculate \( 3(-2) \):
\[ 3(-2) = -6 \]
- Calculate \( 3(-2)^2 \):
\[ 3(-2)^2 = 3 \times 4 = 12 \]
- Calculate \( (-2)^3 \):
\[ (-2)^3 = -8 \]

3. **Combine all the calculated values:**
\[ -64 - 48 - 12 - 6 + 12 - 8 \]

4. **Perform the addition and subtraction step by step:**
- First, combine the negative terms:
\[ -64 - 48 - 12 - 6 - 8 = -138 \]
- Now add the positive term:
\[ -138 + 12 = -126 \]

5. **Final result:**
\[ m^3 - 3m^2 + 3m + 3n + 3n^2 + n^3 = -126 \]

### Conclusion:
The value of the expression is \(-126\).