What is the value of (2+8i)(1-4i)-(3-2i)(6+4i)?

Options

A. 8
B. 26
C. 34
D. 50

Answer

To solve the expression \((2 + 8i)(1 - 4i) - (3 - 2i)(6 + 4i)\), we will follow these steps:

### Step 1: Expand the first product \((2 + 8i)(1 - 4i)\)

Using the distributive property (FOIL method):
\[ (2 + 8i)(1 - 4i) = 2 \cdot 1 + 2 \cdot (-4i) + 8i \cdot 1 + 8i \cdot (-4i) \]
Calculating each term:
\[ = 2 - 8i + 8i - 32i^2 \]

### Step 2: Simplify the first product

Combine like terms:
\[ = 2 - 8i + 8i - 32(-1) \quad (\text{since } i^2 = -1) \]
The \(-8i\) and \(+8i\) cancel out:
\[ = 2 + 32 = 34 \]

### Step 3: Expand the second product \((3 - 2i)(6 + 4i)\)

Using the distributive property (FOIL method):
\[ (3 - 2i)(6 + 4i) = 3 \cdot 6 + 3 \cdot 4i - 2i \cdot 6 - 2i \cdot 4i \]
Calculating each term:
\[ = 18 + 12i - 12i - 8i^2 \]

### Step 4: Simplify the second product

Combine like terms:
\[ = 18 + 12i - 12i - 8(-1) \quad (\text{since } i^2 = -1) \]
The \(12i\) and \(-12i\) cancel out:
\[ = 18 + 8 = 26 \]

### Step 5: Combine the results from both products

Now substitute back into the original expression:
\[ 34 - 26 = 8 \]

### Final Answer

The value of \((2 + 8i)(1 - 4i) - (3 - 2i)(6 + 4i)\) is \(\boxed{8}\).

---