## Factoring the Expression: (x^2 + x + 1)^2 + 2x(x^2 + x + 1) + x^2

This expression looks complex at first glance, but it can be simplified by recognizing a pattern and applying some algebraic techniques.

### Recognizing the Pattern

Notice that the expression resembles a perfect square trinomial. We can rewrite the expression as:

**(x^2 + x + 1)^2 + 2(x^2 + x + 1)(x) + x^2**

This is similar to the expansion of (a + b)^2 = a^2 + 2ab + b^2. Let's substitute:

**a = (x^2 + x + 1)****b = x**

Now, the expression becomes:

**a^2 + 2ab + b^2**

### Applying the Perfect Square Trinomial

We can now factor the expression using the perfect square trinomial pattern:

**(a + b)^2**

Substituting back the values of 'a' and 'b', we get:

**[(x^2 + x + 1) + x]^2**

### Simplifying the Expression

Finally, simplifying the expression:

**(x^2 + 2x + 1)^2**

This is the factored form of the original expression.

### Conclusion

By recognizing the pattern of a perfect square trinomial and applying the corresponding formula, we successfully factored the complex expression: (x^2 + x + 1)^2 + 2x(x^2 + x + 1) + x^2 into its simplified form (x^2 + 2x + 1)^2.