What is the result of the multiplication `(5-6i) (3+3i)` ? (Note: `i = sqrt(-1))`

Options

A. `33-3i`
B. `23-3i`
C. `11-i`
D. `-6`

Answer

To solve the multiplication of the complex numbers \( (5 - 6i)(3 + 3i) \), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps:

### Step-by-Step Solution:

1. **Write down the expression**:
\[ (5 - 6i)(3 + 3i) \]

2. **Apply the distributive property**:
- Multiply the first terms:
\[ 5 \cdot 3 = 15 \]
- Multiply the outer terms:
\[ 5 \cdot 3i = 15i \]
- Multiply the inner terms:
\[ -6i \cdot 3 = -18i \]
- Multiply the last terms:
\[ -6i \cdot 3i = -18i^2 \]

3. **Combine the results**:
\[ 15 + 15i - 18i - 18i^2 \]

4. **Substitute \( i^2 \) with \(-1\)**:
\[ -18i^2 = -18(-1) = 18 \]
So, we can rewrite the expression as:
\[ 15 + 15i - 18i + 18 \]

5. **Combine like terms**:
- Combine the real parts:
\[ 15 + 18 = 33 \]
- Combine the imaginary parts:
\[ 15i - 18i = -3i \]

6. **Final result**:
\[ 33 - 3i \]

### Final Answer:
\[ (5 - 6i)(3 + 3i) = 33 - 3i \]