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 expression \( x^3 - 4x^2 - 7x + 12 \) given that \( x = 4 + 3i \), we will follow these steps:

### Step 1: Substitute \( x \) into the expression
We start by substituting \( x = 4 + 3i \) into the expression \( x^3 - 4x^2 - 7x + 12 \).

### Step 2: Calculate \( x^2 \)
First, we calculate \( x^2 \):
\[ x^2 = (4 + 3i)^2 = 4^2 + 2 \cdot 4 \cdot 3i + (3i)^2 = 16 + 24i + 9i^2 \]
Since \( i^2 = -1 \), we have:
\[ x^2 = 16 + 24i - 9 = 7 + 24i \]

### Step 3: Calculate \( x^3 \)
Next, we calculate \( x^3 \):
\[ x^3 = (4 + 3i)(7 + 24i) = 4 \cdot 7 + 4 \cdot 24i + 3i \cdot 7 + 3i \cdot 24i \]
Calculating each term:
\[ = 28 + 96i + 21i + 72i^2 \]
Again, substituting \( i^2 = -1 \):
\[ = 28 + 96i + 21i - 72 = -44 + 117i \]

### Step 4: Substitute \( x^2 \) and \( x^3 \) into the expression
Now we substitute \( x^2 \) and \( x^3 \) back into the expression:
\[ x^3 - 4x^2 - 7x + 12 = (-44 + 117i) - 4(7 + 24i) - 7(4 + 3i) + 12 \]

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

### Step 6: Calculate \( -7x \)
Calculating \( -7x \):
\[ -7(4 + 3i) = -28 - 21i \]

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

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