If `x=4+3i` (where `i =sqrt(-1)),` then the value of `x ^(3)-4x ^(2)-7x+12` equals:

Options

A. `-88`
B. `48+36i`
C. `-256+12i`
D. `-84`

Answer

To solve the problem, we need to evaluate the expression \( x^3 - 4x^2 - 7x + 12 \) given that \( x = 4 + 3i \).

### Step-by-Step Solution:

1. **Substituting the value of \( x \)**:
\[ x = 4 + 3i \]
We will first calculate \( x^2 \) and \( x^3 \).

2. **Calculating \( x^2 \)**:
\[ x^2 = (4 + 3i)^2 = 4^2 + 2 \cdot 4 \cdot 3i + (3i)^2 \]
\[ = 16 + 24i + 9i^2 \]
Since \( i^2 = -1 \):
\[ = 16 + 24i - 9 = 7 + 24i \]

3. **Calculating \( x^3 \)**:
\[ x^3 = (4 + 3i)(7 + 24i) \]
Using the distributive property:
\[ = 4 \cdot 7 + 4 \cdot 24i + 3i \cdot 7 + 3i \cdot 24i \]
\[ = 28 + 96i + 21i + 72i^2 \]
Again, substituting \( i^2 = -1 \):
\[ = 28 + 96i + 21i - 72 = -44 + 117i \]

4. **Substituting \( x^2 \) and \( x^3 \) into the expression**:
Now we substitute \( x^2 \) and \( x^3 \) into the expression \( x^3 - 4x^2 - 7x + 12 \):
\[ = (-44 + 117i) - 4(7 + 24i) - 7(4 + 3i) + 12 \]

5. **Calculating \( -4x^2 \)**:
\[ -4x^2 = -4(7 + 24i) = -28 - 96i \]

6. **Calculating \( -7x \)**:
\[ -7x = -7(4 + 3i) = -28 - 21i \]

7. **Combining all parts**:
Now we combine all the parts:
\[ = (-44 + 117i) + (-28 - 96i) + (-28 - 21i) + 12 \]
Combining the real parts:
\[ = -44 - 28 - 28 + 12 = -88 \]
Combining the imaginary parts:
\[ = 117i - 96i - 21i = 0i \]

8. **Final Result**:
Therefore, the value of \( x^3 - 4x^2 - 7x + 12 \) is:
\[ \boxed{-88} \]