Which of the following shows `(2+6i)(3i-4)` written as a complex number in the form `a+bi`? (Note: `i^(2)=-1`)

Options

A. `-26-18i`
B. `-18+10i`
C. `10-18i`
D. `30+10i`

Answer

To express the product of the complex numbers \( (2 + 6i)(3i - 4) \) in the form \( a + bi \), we will follow these steps:

### Step 1: Identify the terms
We have:
- \( a = 2 \)
- \( b = 6i \)
- \( c = 3i \)
- \( d = -4 \)

### Step 2: Use the distributive property
We will use the formula for multiplying two binomials:
\[ (a + b)(c + d) = ac + ad + bc + bd \]
Substituting the values:
\[ (2 + 6i)(3i - 4) = 2 \cdot 3i + 2 \cdot (-4) + 6i \cdot 3i + 6i \cdot (-4) \]

### Step 3: Calculate each term
Now we calculate each term:
1. \( 2 \cdot 3i = 6i \)
2. \( 2 \cdot (-4) = -8 \)
3. \( 6i \cdot 3i = 18i^2 \) (Recall that \( i^2 = -1 \))
4. \( 6i \cdot (-4) = -24i \)

### Step 4: Combine the results
Now we combine all the calculated terms:
\[ (2 + 6i)(3i - 4) = 6i - 8 + 18i^2 - 24i \]
Substituting \( i^2 = -1 \):
\[ = 6i - 8 + 18(-1) - 24i \]
\[ = 6i - 8 - 18 - 24i \]

### Step 5: Simplify the expression
Now we combine like terms:
- Real part: \( -8 - 18 = -26 \)
- Imaginary part: \( 6i - 24i = -18i \)

Thus, we have:
\[ (2 + 6i)(3i - 4) = -26 - 18i \]

### Final Result
The expression \( (2 + 6i)(3i - 4) \) in the form \( a + bi \) is:
\[ \boxed{-26 - 18i} \]