The problem is to expand the expression $(x^2 + x + 1)(x^2 - x + 1)$.

1. Problem Description

The problem is to expand the expression $(x^2 + x + 1)(x^2 - x + 1)$.

2. Solution Steps

We can expand the product by multiplying each term of the first polynomial by each term of the second polynomial and then simplifying.

$$(x^2 + x + 1)(x^2 - x + 1) = x^2(x^2 - x + 1) + x(x^2 - x + 1) + 1(x^2 - x + 1)$$

Now we distribute each term:

$$= x^4 - x^3 + x^2 + x^3 - x^2 + x + x^2 - x + 1$$

Combining like terms, we have:

$$= x^4 + (-x^3 + x^3) + (x^2 - x^2 + x^2) + (x - x) + 1$$

$$= x^4 + 0x^3 + x^2 + 0x + 1$$

$$= x^4 + x^2 + 1$$

Alternatively, we can rewrite the expression as

$$(x^2+1+x)(x^2+1-x) = (x^2+1)^2 - x^2$$

Using the formula $(a+b)(a-b) = a^2 - b^2$, where $a = x^2+1$ and $b=x$.

$$(x^2+1)^2 - x^2 = (x^4 + 2x^2 + 1) - x^2 = x^4 + x^2 + 1$$

3. Final Answer

$x^4 + x^2 + 1$