## Expanding Complex Number Expressions: (x - 4 - 5i)(x - 4 + 5i)

This article explores the process of expanding the complex number expression (x - 4 - 5i)(x - 4 + 5i). We'll utilize the distributive property and properties of complex numbers to simplify the expression.

### Understanding the Problem

We're given a product of two binomials with complex coefficients. Our goal is to expand this expression and arrive at a simplified form, ideally eliminating any imaginary components.

### Expanding the Expression

We can expand this expression using the distributive property (also known as FOIL - First, Outer, Inner, Last). Let's break it down step-by-step:

**First:**(x * x) = x²**Outer:**(x * -4) = -4x**Inner:**(-4 * x) = -4x**Last:**(-4 * -4) = 16**First:**(x * 5i) = 5ix**Outer:**(-4 * 5i) = -20i**Inner:**(-5i * x) = -5ix**Last:**(-5i * 5i) = -25i²

Now, we have:

x² - 4x - 4x + 16 + 5ix - 20i - 5ix - 25i²

### Simplifying the Expression

We can simplify this expression by combining like terms and remembering that i² = -1.

**Combining real terms:**x² - 8x + 16**Combining imaginary terms:**5ix - 5ix = 0**Simplifying -25i²:**-25 * (-1) = 25

This leaves us with:

x² - 8x + 16 + 25

Finally, combining the constant terms gives us the simplified expression:

**x² - 8x + 41**

### Conclusion

By expanding and simplifying the complex expression (x - 4 - 5i)(x - 4 + 5i), we arrive at the real quadratic expression x² - 8x + 41. This demonstrates how multiplying complex numbers can result in real-valued expressions.