(8+$$ \sqrt{-5})\left(5-\sqrt{-5}\right) $$=

Given the expression: 

\[
(8 + \sqrt{-5})(5 - \sqrt{-5}),
\]

we need to simplify this product. Since $ \sqrt{-5} $ can be written as $ i\sqrt{5} $ where $ i = \sqrt{-1} $, the expression becomes:

\[
(8 + i\sqrt{5})(5 - i\sqrt{5}).
\]

### Step-by-Step Solution

1. **Use the distributive property (FOIL method):**
   
\[
(8 + i\sqrt{5})(5 - i\sqrt{5}) = 8 \times 5 + 8 \times (-i\sqrt{5}) + i\sqrt{5} \times 5 + i\sqrt{5} \times (-i\sqrt{5}).
\]

2. **Calculate each term:**

   - $ 8 \times 5 = 40 $,
   - $ 8 \times (-i\sqrt{5}) = -8i\sqrt{5} $,
   - $ i\sqrt{5} \times 5 = 5i\sqrt{5} $,
   - $ i\sqrt{5} \times (-i\sqrt{5}) = -i^2 \times (\sqrt{5})^2 = -(-1) \times 5 = 5 $ (since $ i^2 = -1 $ and $ (\sqrt{5})^2 = 5 $).

3. **Combine all the terms:**

\[
40 - 8i\sqrt{5} + 5i\sqrt{5} + 5.
\]

4. **Combine like terms:**

   - Real part: $ 40 + 5 = 45 $,
   - Imaginary part: $ -8i\sqrt{5} + 5i\sqrt{5} = -3i\sqrt{5} $.

So, the simplified expression is:

\[
45 - 3i\sqrt{5}.
\]

### Final Answer

\[
(8 + \sqrt{-5})(5 - \sqrt{-5}) = 45 - 3i\sqrt{5}.
\]

Would you like more details or have any questions?

### Follow-up Questions

1. What happens if you multiply other complex conjugate pairs?
2. How do you express other negative square roots in terms of $ i $?
3. Can you expand $ (a + bi)(c - di) $ in general?
4. What are some real-life applications of complex numbers?
5. How do you find the modulus of a complex number like $ 45 - 3i\sqrt{5} $?

### Tip

When multiplying complex numbers, always keep track of $ i^2 = -1 $ to simplify your expressions correctly!

Explore the process of simplifying complex number expressions like (8 + √-5)(5 - √-5) using the distributive property and imaginary numbers. This detailed solution demonstrates key concepts of algebra, including the use of i^2 = -1 and combining like terms. Suitable for high school students in Grades 9-12.

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