## Proof of (a+b)(c+d)=ac+ad+bc+bd

This equation is a fundamental concept in algebra, often referred to as the **distributive property**. It states that the product of two binomials can be expanded as the sum of four terms.

Here's a breakdown of the proof:

### 1. Understanding the terms:

**(a + b)**and**(c + d)**are two binomials, each containing two terms.**ac**,**ad**,**bc**, and**bd**are the four terms we aim to obtain after the expansion.

### 2. Applying the distributive property:

We start by distributing the first term of the first binomial (a) to both terms of the second binomial:

**a(c + d) = ac + ad**

Next, we distribute the second term of the first binomial (b) to both terms of the second binomial:

**b(c + d) = bc + bd**

### 3. Combining the results:

Finally, we add the two results from step 2 to get the complete expansion:

**(a + b)(c + d) = ac + ad + bc + bd**

### Example:

Let's illustrate this with an example:

Suppose we have:

- a = 2
- b = 3
- c = 4
- d = 5

Using the equation, we can find the product:

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

Simplifying the equation:

5 * 9 = 8 + 10 + 12 + 15

45 = 45

Therefore, we have verified that the equation (a + b)(c + d) = ac + ad + bc + bd holds true.

### Conclusion:

This proof demonstrates the distributive property in action, allowing us to expand the product of two binomials into a sum of four terms. This principle is fundamental in algebra and forms the basis for many other algebraic manipulations and calculations.