## Expanding the Expression: (x^2 - x + 1)(x^2 + x + 1)

This expression involves multiplying two quadratic polynomials. We can expand it using the distributive property (also known as FOIL) or by recognizing a pattern.

### Using the Distributive Property (FOIL)

**FOIL** stands for **First, Outer, Inner, Last**. It's a mnemonic device to remember all the terms that need to be multiplied when expanding two binomials.

**First:**Multiply the first terms of each binomial: x² * x² = x⁴**Outer:**Multiply the outer terms of each binomial: x² * x = x³**Inner:**Multiply the inner terms of each binomial: -x * x² = -x³**Last:**Multiply the last terms of each binomial: -x * x = -x²

Now we need to multiply 1 from the first binomial by each term in the second binomial:

- 1 * x² = x²
- 1 * x = x
- 1 * 1 = 1

Combining all the terms:

x⁴ + x³ - x³ - x² + x² + x + 1

Simplifying by combining like terms:

**x⁴ + x + 1**

### Using a Pattern Recognition

Notice that the two binomials are conjugates of each other, meaning they have the same terms but opposite signs in the middle. This leads to a pattern:

**(a + b)(a - b) = a² - b²**

In our case, a = x² and b = x - 1.

Applying the pattern:

(x² - x + 1)(x² + x + 1) = (x²)² - (x - 1)²

Expanding the square:

= x⁴ - (x² - 2x + 1)

Simplifying:

= x⁴ - x² + 2x - 1

= **x⁴ + x + 1**

### Conclusion

Both methods lead to the same simplified expression: **x⁴ + x + 1**. Recognizing the pattern can make the expansion faster, but understanding the distributive property is crucial for more complex expressions.