Which of the following complex numbers is equal to (5 + 12i) − (`9i^2` − 6i), for `i = sqrt(−1)` ?

Options

A. `−14 − 18i`
B. `−4 − 6i`
C. 4 + 6i
D. 14 + 18i

Explanation

To solve the problem, we need to simplify the expression \( (5 + 12i) - (9i^2 - 6i) \) where \( i = \sqrt{-1} \).

### Step-by-Step Solution:

1. **Identify the expression**:
We start with the expression:
\[ (5 + 12i) - (9i^2 - 6i) \]

2. **Substitute \( i^2 \)**:
We know that \( i^2 = -1 \). Therefore, we can replace \( i^2 \) in the expression:
\[ (5 + 12i) - (9(-1) - 6i) \]

3. **Simplify the expression inside the parentheses**:
This becomes:
\[ (5 + 12i) - (-9 - 6i) \]
Simplifying further, we have:
\[ (5 + 12i) + 9 + 6i \]

4. **Combine like terms**:
Now, we combine the real parts and the imaginary parts:
- Real parts: \( 5 + 9 = 14 \)
- Imaginary parts: \( 12i + 6i = 18i \)

Thus, we get:
\[ 14 + 18i \]

5. **Final result**:
The simplified expression is:
\[ 14 + 18i \]

### Conclusion:
The complex number equal to \( (5 + 12i) - (9i^2 - 6i) \) is \( 14 + 18i \), which corresponds to option D.