## Exploring the Expansion of (x+1)(x+2)(x+3)(x+4)(x+5)

This seemingly simple expression hides a fascinating pattern and can be expanded in multiple ways. Let's delve into the different approaches and explore the resulting polynomial.

### Direct Expansion:

The most straightforward approach is to expand the expression term by term.

**Multiply the first two factors:**(x+1)(x+2) = x² + 3x + 2**Multiply the result by the third factor:**(x² + 3x + 2)(x+3) = x³ + 6x² + 11x + 6**Continue multiplying by the remaining factors:**(x³ + 6x² + 11x + 6)(x+4) = x⁴ + 10x³ + 35x² + 50x + 24**Finally, multiply by the last factor:**(x⁴ + 10x³ + 35x² + 50x + 24)(x+5) =**x⁵ + 15x⁴ + 85x³ + 225x² + 274x + 120**

While this method works, it can become tedious for larger expressions.

### Utilizing Patterns:

Observe that the expression represents the product of five consecutive integers increased by 1. This leads to interesting patterns:

**The constant term:**The product of the constant terms in each factor (1*2*3*4*5) equals 120. This is always the product of the consecutive integers.**The leading coefficient:**The leading coefficient is always 1, as the highest power of x is obtained by multiplying the 'x' terms from each factor.**The coefficients of other terms:**The coefficients of the other terms are related to the sum of products of the consecutive integers. For example, the coefficient of x⁴ is the sum of all possible products of four of the consecutive integers (1*2*3*4 + 1*2*3*5 + 1*2*4*5 + 1*3*4*5 + 2*3*4*5), which equals 15.

### The Binomial Theorem Approach:

While less obvious, the Binomial Theorem can be used to expand this expression. We can rewrite the expression as:

(x+1)(x+2)(x+3)(x+4)(x+5) = (x+5) * (x+4) * (x+3) * (x+2) * (x+1)

Now, let's consider the product of the last four factors:

(x+4) * (x+3) * (x+2) * (x+1) = (x⁴ + 10x³ + 35x² + 50x + 24)

This resembles the form of the Binomial Theorem expansion. We can rewrite it as:

(x + 5) * (x⁴ + 10x³ + 35x² + 50x + 24) = (x+5) * (x⁴ + (10*5)x³ + (35*5²)x² + (50*5³)x + (24*5⁴))

Finally, expanding this product gives us the same result as the direct expansion:
**x⁵ + 15x⁴ + 85x³ + 225x² + 274x + 120**

### Conclusion:

Expanding (x+1)(x+2)(x+3)(x+4)(x+5) reveals a fascinating polynomial with interesting patterns. While direct expansion is the most straightforward method, utilizing patterns and even the Binomial Theorem offer alternative approaches to understanding this intriguing expression.