Understanding the Distributive Property with (x+4)(x+5)
The distributive property is a fundamental concept in algebra, allowing us to simplify expressions involving multiplication. Let's explore how it works with the example of (x+4)(x+5).
What is the Distributive Property?
The distributive property states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products. In simpler terms, a(b + c) = ab + ac.
Applying the Distributive Property to (x+4)(x+5)

Expand the first term: We start by distributing the 'x' from the first binomial to each term in the second binomial:
 x(x+5) = x² + 5x

Expand the second term: Now, we distribute the '4' from the first binomial to each term in the second binomial:
 4(x+5) = 4x + 20

Combine the results: Finally, we add the two results from steps 1 and 2:
 x² + 5x + 4x + 20

Simplify: Combining like terms, we get the simplified expression:
 x² + 9x + 20
Visualizing the Distributive Property
The distributive property can be visualized as a "box method". Imagine a rectangle divided into four smaller rectangles.
 Top Left: x * x = x²
 Top Right: x * 5 = 5x
 Bottom Left: 4 * x = 4x
 Bottom Right: 4 * 5 = 20
Adding all the terms inside the boxes gives us the expanded expression: x² + 5x + 4x + 20, which simplifies to x² + 9x + 20.
Summary
Using the distributive property, we have successfully expanded and simplified the expression (x+4)(x+5) to x² + 9x + 20. This process demonstrates how the distributive property simplifies multiplication involving sums, making it an essential tool in algebra.