Using the Distributive Property to Expand (x+1)(x+3)
The distributive property is a fundamental concept in algebra that allows us to expand expressions involving multiplication. It states that multiplying a sum by a number is the same as multiplying each term in the sum by that number. In simpler terms:
a(b + c) = ab + ac
We can use this property to expand the expression (x+1)(x+3).
Steps to Expand (x+1)(x+3)

Treat (x+1) as a single term. We can rewrite the expression as: (x+1)(x+3) = (x+1) * (x+3)

Apply the distributive property. We multiply each term in the first set of parentheses by each term in the second set of parentheses.
(x+1) * (x+3) = x * (x+3) + 1 * (x+3)

Simplify each multiplication.
x * (x+3) + 1 * (x+3) = x^2 + 3x + x + 3

Combine like terms.
x^2 + 3x + x + 3 = x^2 + 4x + 3
Therefore, the expanded form of (x+1)(x+3) is x^2 + 4x + 3.
Visualizing the Distributive Property
It can be helpful to visualize the distributive property using a diagram. Imagine a rectangle with length (x+1) and width (x+3).
[Image of a rectangle with length (x+1) and width (x+3) divided into four smaller rectangles]
The area of the rectangle is the product of its length and width: (x+1)(x+3). We can find the area of the whole rectangle by adding the areas of the four smaller rectangles:
 x * x = x^2
 x * 3 = 3x
 1 * x = x
 1 * 3 = 3
Adding these areas gives us x^2 + 3x + x + 3, which is the same result we obtained using the distributive property.
Conclusion
The distributive property is a powerful tool for simplifying and expanding algebraic expressions. By understanding and applying this property, we can efficiently manipulate expressions and solve equations in a variety of contexts.