## Factoring the Expression (x^2 + x + 1)(x^2 + x + 2) - 12

This expression appears complex, but we can simplify it through strategic factoring.

### 1. Recognizing the Pattern

Notice that the terms within the parentheses share a common pattern: both are quadratic expressions of the form x^2 + x + c, where c is a constant. This pattern is key to simplifying the expression.

### 2. Substitution for Easier Manipulation

Let's introduce a temporary variable, say **y**, to represent the common expression:

**y = x^2 + x**

Now, our expression becomes:

*(y + 1)(y + 2) - 12*

### 3. Expanding and Simplifying

Expanding the expression, we get:

- y^2 + 3y + 2 - 12 = y^2 + 3y - 10

### 4. Factoring the Quadratic

We now have a simple quadratic equation. We can factor it into:

- (y + 5)(y - 2)

### 5. Substituting Back

Finally, substitute back **x^2 + x** for **y**:

- (x^2 + x + 5)(x^2 + x - 2)

### 6. Further Factoring (Optional)

The second factor can be factored further:

- (x^2 + x + 5)(x + 2)(x - 1)

### Conclusion

Therefore, the fully factored form of the expression (x^2 + x + 1)(x^2 + x + 2) - 12 is **(x^2 + x + 5)(x + 2)(x - 1)**. This method demonstrates how recognizing patterns and strategic substitution can simplify seemingly complex algebraic expressions.