Find `(2+4i)(3-2i)+(-3i)(5i)`

Options

A. `-17+8i`
B. `13+8i`
C. `13+16i`
D. `29+8i`

Answer

To solve the expression \((2 + 4i)(3 - 2i) + (-3i)(5i)\), we will follow these steps:

### Step 1: Expand the first part \((2 + 4i)(3 - 2i)\)

Using the distributive property (also known as the FOIL method for binomials):

\[ (2 + 4i)(3 - 2i) = 2 \cdot 3 + 2 \cdot (-2i) + 4i \cdot 3 + 4i \cdot (-2i) \]

Calculating each term:

- \(2 \cdot 3 = 6\)
- \(2 \cdot (-2i) = -4i\)
- \(4i \cdot 3 = 12i\)
- \(4i \cdot (-2i) = -8i^2\)

Now substituting \(i^2 = -1\):

\(-8i^2 = -8(-1) = 8\)

Combining all these results:

\[ 6 - 4i + 12i + 8 = 6 + 8 + (-4i + 12i) = 14 + 8i \]

### Step 2: Calculate the second part \((-3i)(5i)\)

\[ (-3i)(5i) = -15i^2 \]

Again substituting \(i^2 = -1\):

\(-15i^2 = -15(-1) = 15\)

### Step 3: Combine the results from Step 1 and Step 2

Now we add the results from both parts:

\[ (14 + 8i) + 15 = 14 + 15 + 8i = 29 + 8i \]

### Final Answer

Thus, the final result is:

\[ \boxed{29 + 8i} \]

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