The Curious Case of (a+1)(a+2)(a+3)(a+4)+1
The expression (a+1)(a+2)(a+3)(a+4)+1 might seem like a simple algebraic expression, but it holds a fascinating property. Let's delve into this property and explore why it's so intriguing.
The Hidden Pattern:
The remarkable thing about this expression is that it can always be factored into a product of two quadratic expressions, regardless of the value of 'a'. Here's how it works:

Expand the expression: The initial step is to expand the product of the first four terms: (a+1)(a+2)(a+3)(a+4) = a⁴ + 10a³ + 35a² + 50a + 24

Add 1 and rearrange: Now, we add 1 and rearrange the terms: a⁴ + 10a³ + 35a² + 50a + 25 = a⁴ + 10a³ + 35a² + 50a + 25

Factor by grouping: We can now factor the expression by grouping: (a⁴ + 5a³ + 5a² ) + (5a³ + 25a² + 25a) + (5a² + 25a + 25) = a²(a² + 5a + 5) + 5a(a² + 5a + 5) + 5(a² + 5a + 5)

Final factorization: This gives us the final factored form: (a² + 5a + 5)(a² + 5a + 5) = (a² + 5a + 5)²
The Significance:
This factorization reveals that the expression (a+1)(a+2)(a+3)(a+4)+1 is always a perfect square. This is a surprising result, as the original expression doesn't readily suggest this property.
The fact that it always results in a perfect square regardless of the value of 'a' showcases a beautiful connection between algebra and the concept of perfect squares. It also highlights the importance of recognizing patterns and using creative algebraic manipulation to uncover hidden relationships.
Further Exploration:
This property can be explored further. For example, one could investigate similar expressions with different numbers of terms. Would they also exhibit this perfect square property? This exploration leads to deeper insights into the fascinating world of algebraic identities and their applications.