Expanding the Expression (x+1)(x+2)(x+3)(x+4)
This article explores the expansion of the expression (x+1)(x+2)(x+3)(x+4). While it might seem daunting at first, we can break down the problem into manageable steps.
StepbyStep Expansion:

Pairwise Multiplication: Begin by multiplying the first two factors and the last two factors:
 (x+1)(x+2) = x² + 3x + 2
 (x+3)(x+4) = x² + 7x + 12

Expanding the Remaining Expression: Now, we need to multiply the results from step 1:
 (x² + 3x + 2)(x² + 7x + 12)

Distribution: To multiply these two trinomials, we can distribute each term of the first trinomial across the second trinomial:
 x²(x² + 7x + 12) + 3x(x² + 7x + 12) + 2(x² + 7x + 12)

Simplifying: Now, expand and combine like terms:
 x⁴ + 7x³ + 12x² + 3x³ + 21x² + 36x + 2x² + 14x + 24

Final Result: Finally, combine all the terms to get the simplified form:
 x⁴ + 10x³ + 35x² + 50x + 24
Understanding the Expansion:
Expanding this expression demonstrates the concept of polynomial multiplication. Each step involves distributing terms and combining like terms, resulting in a polynomial with a higher degree than the original factors. This process is essential in various mathematical applications, including solving equations, factoring, and analyzing functions.
Applications:
 Solving Equations: When the expression is set equal to zero, the resulting equation can be solved to find the roots or solutions.
 Factoring: The expansion helps understand how the original expression can be factored back into its individual factors.
 Calculus: Understanding the expanded form is crucial for calculating derivatives and integrals of polynomial functions.
In conclusion, expanding (x+1)(x+2)(x+3)(x+4) involves systematic multiplication and simplification. This process reveals the powerful concepts of polynomial multiplication, essential for various mathematical applications.