## Multiplying Complex Numbers: (x + 1 + 3i)(x + 1 - 3i)

This expression involves multiplying two complex numbers, which can be approached using the **FOIL method** (First, Outer, Inner, Last) or by recognizing the pattern of a **difference of squares**. Let's explore both approaches.

### Using FOIL Method

**First:**(x)(x) =**x²****Outer:**(x)(-3i) =**-3ix****Inner:**(1)(x) =**x****Last:**(1)(-3i) =**-3i****Middle terms:**We also have (3i)(x) =**3ix**and (3i)(-3i) =**-9i²**

Combining all the terms, we get:

x² - 3ix + x - 3i + 3ix - 9i²

Since **i² = -1**, we can simplify further:

x² + x - 3i + 3ix + 9

Finally, combining real and imaginary terms:

**x² + x + 9**

### Using Difference of Squares

We can recognize the given expression as a difference of squares:

(x + 1 + 3i)(x + 1 - 3i) = [(x + 1) + 3i][(x + 1) - 3i]

The difference of squares pattern states: (a + b)(a - b) = a² - b²

Applying this pattern:

[(x + 1) + 3i][(x + 1) - 3i] = (x + 1)² - (3i)²

Expanding and simplifying:

x² + 2x + 1 - 9i² = x² + 2x + 1 + 9

**Therefore, (x + 1 + 3i)(x + 1 - 3i) simplifies to x² + 2x + 10.**

### Key Points

- This problem demonstrates how to multiply complex numbers using the FOIL method and the difference of squares pattern.
- The result of multiplying these complex conjugates is always a real number.
- Understanding complex number operations is essential in various fields, including mathematics, physics, and engineering.