If `a-b=3, a+b+x=2` , then the value of `(a-b)[x^(3)+3(a+b)x^(2)+3x(a+b)^(2)+(a+b)^(3)]` is

Options

A. `84`
B. `48`
C. `32`
D. `24`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
- \( a - b = 3 \)
- \( a + b + x = 2 \)

2. **Express \( x \) in terms of \( a \) and \( b \):**
From the second equation, we can isolate \( x \):
\[ x = 2 - (a + b) \]

3. **Substitute \( a + b \) in terms of \( a - b \):**
We know from the first equation that \( a - b = 3 \). We can express \( a + b \) in terms of \( a - b \):
- Adding the two equations:
\[ (a - b) + (a + b) = 3 + (a + b) \]
This simplifies to:
\[ 2a = 3 + (a + b) \]
- Therefore, \( a + b = 2a - 3 \).

4. **Now, we can substitute \( a + b \) back into the equation for \( x \):**
\[ x = 2 - (2a - 3) = 5 - 2a \]

5. **Now we need to evaluate the expression:**
\[ (a - b)[x^3 + 3(a + b)x^2 + 3x(a + b)^2 + (a + b)^3] \]
We can recognize that the expression inside the brackets is of the form \((x + (a + b))^3\) based on the binomial expansion:
\[ x^3 + 3(a + b)x^2 + 3x(a + b)^2 + (a + b)^3 = (x + (a + b))^3 \]

6. **Substituting \( a + b \):**
\[ (x + (a + b)) = (5 - 2a + (2a - 3)) = 2 \]

7. **Now we can calculate the cube:**
\[ (x + (a + b))^3 = 2^3 = 8 \]

8. **Finally, substitute back into the original expression:**
\[ (a - b) \cdot 8 = 3 \cdot 8 = 24 \]

Thus, the final value is:
\[ \boxed{24} \]