If `i=sqrt(-1)`, then `(7+5i)(-2-6i)=`

Options

A. 16
B. `-44`
C. `16-52i`
D. `-44-52i`

Answer

To solve the expression \((7 + 5i)(-2 - 6i)\), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps:

### Step 1: Distribute the terms
We will multiply each term in the first binomial by each term in the second binomial.

\[ (7 + 5i)(-2 - 6i) = 7 \cdot (-2) + 7 \cdot (-6i) + 5i \cdot (-2) + 5i \cdot (-6i) \]

### Step 2: Calculate each product
Now we will calculate each of the products:

1. \(7 \cdot (-2) = -14\)
2. \(7 \cdot (-6i) = -42i\)
3. \(5i \cdot (-2) = -10i\)
4. \(5i \cdot (-6i) = -30i^2\)

### Step 3: Substitute \(i^2\) with \(-1\)
Since \(i^2 = -1\), we can substitute this into our expression:

\[ -30i^2 = -30(-1) = 30 \]

### Step 4: Combine all the terms
Now, we will combine all the calculated terms:

\[ -14 - 42i - 10i + 30 \]

Combine the real parts and the imaginary parts:

Real parts: \(-14 + 30 = 16\)

Imaginary parts: \(-42i - 10i = -52i\)

### Step 5: Write the final result
Putting it all together, we get:

\[ 16 - 52i \]

Thus, the final answer is:

\[ (7 + 5i)(-2 - 6i) = 16 - 52i \]

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