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The problem involves multiplying two complex numbers: \( (2 + 7i) \) and \( (5 - 9i) \). The goal is to simplify the expression in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Let's solve it step-by-step using the distributive property (also known as FOIL method for binomials):

\[
(2 + 7i)(5 - 9i) = 2(5) + 2(-9i) + 7i(5) + 7i(-9i)
\]

1. \( 2(5) = 10 \)
2. \( 2(-9i) = -18i \)
3. \( 7i(5) = 35i \)
4. \( 7i(-9i) = -63i^2 \)

Since \( i^2 = -1 \), the last term becomes:

\[
-63i^2 = -63(-1) = 63
\]

Now, add all the terms together:

\[
10 + (-18i) + 35i + 63
\]

Combine like terms:

1. Real part: \( 10 + 63 = 73 \)
2. Imaginary part: \( -18i + 35i = 17i \)

So, the result is:

\[
73 + 17i
\]

Thus, the simplified answer is \( 73 + 17i \).

Would you like further details or have any questions? Here are five related questions:

1. How do you multiply complex numbers using the FOIL method?
2. Why does \( i^2 = -1 \)?
3. Can you explain why the real and imaginary parts are combined separately?
4. How would you divide complex numbers?
5. What happens if both complex numbers have negative real parts?

**Tip:** Always remember to use \( i^2 = -1 \) when simplifying complex number multiplications!

Learn how to multiply complex numbers (2 + 7i) and (5 - 9i) using the FOIL method. This problem includes key concepts such as complex number multiplication, imaginary unit properties, and simplifying the expression into the form a + bi. Suitable for high school students.

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