Multiply:

Question

left-parenthesis 2 minus 6 i right-parenthesis left-parenthesis negative 4 plus 3 i right-parenthesis

Answer options with 4 options
A.
negative 26 minus 30 i

B.
negative 26 plus 30 i

C.
10 minus 30 i

D.
10 plus 30 i

Answer 1

To multiply the complex numbers \( (2 - 6i)(-4 + 3i) \), we can use the distributive property (also known as the FOIL method).

Let's calculate it step-by-step:

1. First terms: \[ 2 \cdot (-4) = -8 \]

2. Outer terms: \[ 2 \cdot 3i = 6i \]

3. Inner terms: \[ -6i \cdot (-4) = 24i \]

4. Last terms: \[ -6i \cdot 3i = -18i^2 \] Since \( i^2 = -1 \), we have: \[ -18i^2 = -18(-1) = 18 \]

Now, combining all these results: \[ -8 + 6i + 24i + 18 \]

Now, combine the real parts and the imaginary parts: \[ (-8 + 18) + (6i + 24i) = 10 + 30i \]

Thus, the result of the multiplication is: \[ 10 + 30i \]

So the correct answer is option D. \( 10 + 30i \).