To expand the expression \((x - 1)(x^2 + x + 1)\), we apply the distributive property (also known as the FOIL method for binomials) by multiplying each term in the first polynomial by each term in the second polynomial:

\[
(x - 1)(x^2 + x + 1)
\]

### Step 1: Distribute \(x\) over \((x^2 + x + 1)\)
\[
x \cdot (x^2 + x + 1) = x^3 + x^2 + x
\]

### Step 2: Distribute \(-1\) over \((x^2 + x + 1)\)
\[
-1 \cdot (x^2 + x + 1) = -x^2 - x - 1
\]

### Step 3: Combine like terms
Now, add the results from the previous steps:

\[
x^3 + x^2 + x - x^2 - x - 1
\]

Combine the like terms:

\[
x^3 + (x^2 - x^2) + (x - x) - 1 = x^3 - 1
\]

### Final Result
The expanded form of \((x - 1)(x^2 + x + 1)\) is:

\[
x^3 - 1
\]

Would you like more details or have any questions?

Here are 8 related questions:

1. What happens if you expand \((x + 1)(x^2 - x + 1)\)?
2. Can you factorize the expression \(x^3 - 1\) back into its original form?
3. How would the expansion change if the expression were \((x + 2)(x^2 + 3x + 1)\)?
4. What is the general formula for expanding \((x - a)(x^2 + ax + a^2)\)?
5. Can you explain the significance of the result \(x^3 - 1\) in terms of roots?
6. How would you expand \((x^2 - 2x + 3)(x^2 + x + 1)\)?
7. What is the relationship between the expanded form and the roots of the polynomial \(x^3 - 1\)?
8. How do you approach expanding a product of polynomials with more than two factors?

**Tip:** When expanding polynomials, ensure that you multiply each term in the first polynomial by every term in the second polynomial to avoid missing any terms.

Learn how to expand the expression (x - 1)(x^2 + x + 1) using the distributive property (FOIL method). This detailed solution covers polynomial expansion, including combining like terms and the step-by-step process. Suitable for high school students studying algebra and polynomials.

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