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To simplify the given complex expression \(\frac{1 + 8i}{-5 - 9i}\) into the form \(a + bi\), we need to multiply both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of \(-5 - 9i\) is \(-5 + 9i\).

### Steps:
1. Multiply both the numerator and denominator by \(-5 + 9i\):
   \[
   \frac{(1 + 8i)(-5 + 9i)}{(-5 - 9i)(-5 + 9i)}
   \]

2. Simplify the denominator:
   \[
   (-5 - 9i)(-5 + 9i) = (-5)^2 - (9i)^2 = 25 - 81(-1) = 25 + 81 = 106
   \]

3. Now simplify the numerator by using the distributive property (FOIL):
   \[
   (1 + 8i)(-5 + 9i) = 1(-5) + 1(9i) + 8i(-5) + 8i(9i)
   \]
   \[
   = -5 + 9i - 40i + 72i^2
   \]
   Since \(i^2 = -1\), the expression becomes:
   \[
   -5 + 9i - 40i + 72(-1) = -5 + 9i - 40i - 72 = -5 - 72 + (9i - 40i)
   \]
   \[
   = -77 - 31i
   \]

4. Now, the expression becomes:
   \[
   \frac{-77 - 31i}{106}
   \]

5. Separate the real and imaginary parts:
   \[
   = \frac{-77}{106} + \frac{-31}{106}i
   \]

So the simplified form is:
\[
\boxed{-\frac{77}{106} - \frac{31}{106}i}
\]

Would you like more details on any step or have further questions?

Here are 5 related questions:
1. What is the purpose of multiplying by the conjugate in complex division?
2. How can you verify the correctness of the final simplified form?
3. What are the general properties of imaginary numbers used in this solution?
4. How does multiplying complex numbers follow the distributive property?
5. How would this process differ if the denominator was already in its conjugate form?

**Tip:** Always remember that when dividing complex numbers, multiplying by the conjugate simplifies the denominator into a real number, allowing easier expression in the \(a + bi\) form.

Express as a complex number in simplest a+bi form: (1 + 8i) / (-5 - 9i)

Learn how to simplify the complex number (1 + 8i) / (-5 - 9i) into its a + bi form. This problem uses concepts such as complex conjugates and the FOIL method for multiplication. Ideal for high school students in Grades 10-12.

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