## Factoring and Exploring the Expression (x+1)(x-1)(x-3)(x-5) + 12

This expression, (x+1)(x-1)(x-3)(x-5) + 12, presents a unique opportunity to explore concepts like factoring, symmetry, and the relationship between roots and coefficients. Let's break it down step by step.

### Understanding the Structure

At first glance, the expression looks complicated. However, the first four terms are a product of four binomials. This structure suggests a pattern and potential for simplification.

### Recognizing the Difference of Squares

Notice that the first two binomials (x+1) and (x-1) are in the form of a difference of squares: (a+b)(a-b) = a² - b². Applying this pattern, we can rewrite the expression as:

(x² - 1)(x-3)(x-5) + 12

### Further Simplification

Now, we can expand the remaining terms:

(x² - 1)(x² - 8x + 15) + 12

Multiplying the two quadratic expressions, we get:

x⁴ - 8x³ + 15x² - x² + 8x - 15 + 12

Combining like terms, we obtain the simplified expression:

**x⁴ - 8x³ + 14x² + 8x - 3**

### Exploring the Roots

The simplified expression is a quartic polynomial. Finding its roots (values of x that make the expression equal to zero) is a bit more challenging than for quadratics. However, we can observe some key points:

**Symmetry:**The expression exhibits a symmetrical pattern in its coefficients. The terms with even powers of x (x⁴ and 14x²) have positive coefficients, while the terms with odd powers (x³ and x) have negative coefficients. This symmetry hints at a possible factorization.**Relation to the Original Form:**The original expression (x+1)(x-1)(x-3)(x-5) + 12 suggests that it might have roots at x = -1, 1, 3, and 5. However, the constant term (12) indicates that these values might not be exact roots.

### Conclusion

By simplifying the expression and analyzing its structure, we gain insights into its behavior and potential factorization. Further exploration could involve using numerical methods or factorization techniques to find the exact roots of the polynomial. The symmetrical nature of the expression provides a starting point for finding these roots and understanding its behavior.