Question

Simplify:



left-parenthesis 3 plus 2 i right-parenthesis left-parenthesis 2 plus 5 i right-parenthesis

Answer options with 4 options
A.
negative 4 plus 19 i

B.
negative 4 plus 15 i

C.
16 plus 19 i

D.
16 plus 15 i

Answer 1

To simplify the expression \((3 + 2i)(2 + 5i)\), we will use the distributive property (also known as the FOIL method for binomials).

First, distribute each term in the first binomial across each term in the second binomial:

\[
(3 + 2i)(2 + 5i) = 3 \cdot 2 + 3 \cdot 5i + 2i \cdot 2 + 2i \cdot 5i
\]

Calculating each term:

1. \(3 \cdot 2 = 6\)
2. \(3 \cdot 5i = 15i\)
3. \(2i \cdot 2 = 4i\)
4. \(2i \cdot 5i = 10i^2\)

Recall that \(i^2 = -1\), so we can rewrite \(10i^2\) as:

\[
10i^2 = 10(-1) = -10
\]

Now combine all the terms:

\[
6 + 15i + 4i - 10
\]

Combine the real and imaginary parts:

\[
(6 - 10) + (15i + 4i) = -4 + 19i
\]

Thus, the simplified result is:

\[
\boxed{-4 + 19i}
\]

The correct answer is **A. \(-4 + 19i\)**.