Perform the indicated operation & simplify. Express the answer as a complex number.

(8-12i)(-5-7i)

To perform the operation \((8 - 12i)(-5 - 7i)\) and express the answer as a complex number, we'll use the distributive property (FOIL method):

\[
(8 - 12i)(-5 - 7i) = 8(-5) + 8(-7i) - 12i(-5) - 12i(-7i)
\]

Now, let's calculate each term:

1. \(8(-5) = -40\)
2. \(8(-7i) = -56i\)
3. \(-12i(-5) = 60i\)
4. \(-12i(-7i)\)

The term \(-12i(-7i)\) simplifies as follows:

\[
-12i(-7i) = 84i^2
\]

Since \(i^2 = -1\), this becomes:

\[
84i^2 = 84(-1) = -84
\]

Now, combine all the terms:

\[
-40 - 56i + 60i - 84
\]

Combine the real and imaginary parts:

- Real part: \(-40 - 84 = -124\)
- Imaginary part: \(-56i + 60i = 4i\)

So, the simplified form of the expression is:

\[
\boxed{-124 + 4i}
\]

Would you like more details or have any questions about the process? Here are some related questions for further practice:

1. Simplify \((3 + 2i)(4 - 5i)\).
2. Find the product of \((7 - i)(3 + 4i)\).
3. Simplify \((2 - 3i)^2\).
4. Perform the operation and simplify \((5 + 6i)(-2 + 9i)\).
5. Multiply \((1 - 2i)(-3 + 7i)\) and simplify.

**Tip:** When multiplying complex numbers, always remember to apply the distributive property and simplify using \(i^2 = -1\).

Learn how to perform and simplify the multiplication of complex numbers using the distributive property and imaginary unit properties. Detailed step-by-step solution provided for the expression (8-12i)(-5-7i). Suitable for advanced high school students and anyone interested in complex number arithmetic.

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