## Exploring the Expression (x+1)(x+3)(x+5)(x+7)+15

This article delves into the fascinating algebraic expression **(x+1)(x+3)(x+5)(x+7)+15**. We will explore its properties, potential factorization, and interesting patterns that emerge.

### Understanding the Structure

At first glance, the expression appears complex. However, we can break it down:

**Four linear factors:**(x+1), (x+3), (x+5), and (x+7) are all linear expressions, representing straight lines when graphed.**Constant term:**The addition of 15 introduces a constant value, impacting the overall behavior of the expression.

### Factoring Strategies

To simplify the expression, we aim to factor it. Let's try a couple of approaches:

**1. Grouping and Rearrangement**

- We can group the first two and the last two factors: [(x+1)(x+3)][(x+5)(x+7)] + 15.
- Expanding each group: (x² + 4x + 3)(x² + 12x + 35) + 15.
- Now, it's not immediately obvious how to factor further.

**2. Recognizing a Pattern**

- Notice the constant terms in the factors: 1, 3, 5, and 7. These are consecutive odd numbers.
- The constant term 15 is the product of 3 and 5 (two of the numbers in the sequence).
- This suggests a potential pattern that might lead to simplification.

### The Key Observation

We can rewrite the expression in a more insightful way:

(x+1)(x+3)(x+5)(x+7)+15 = [(x+1)(x+7)][(x+3)(x+5)] + 15

Now, notice that:

- (x+1)(x+7) = x² + 8x + 7
- (x+3)(x+5) = x² + 8x + 15

The expression becomes:

(x² + 8x + 7)(x² + 8x + 15) + 15

### Final Factorization

Let's make a substitution to simplify the expression further. Let y = x² + 8x:

(y + 7)(y + 15) + 15 = y² + 22y + 105 + 15 = y² + 22y + 120

Now, we can factor the quadratic:

y² + 22y + 120 = (y + 10)(y + 12)

Substituting back y = x² + 8x:

(x² + 8x + 10)(x² + 8x + 12)

This is the completely factored form of the expression.

### Conclusion

The initial seemingly complex expression **(x+1)(x+3)(x+5)(x+7)+15** reveals a fascinating pattern and simplifies to **(x² + 8x + 10)(x² + 8x + 12)** through careful manipulation and factorization. This exploration highlights the importance of recognizing patterns and utilizing strategic algebraic techniques to simplify expressions.