## Expanding and Simplifying Algebraic Expressions

This article explores the given algebraic equation:

**(a + b + c + d)(a - b - c + d) = (a - b + c - d)(a + b - c - d)**

We will prove that this equation holds true for any values of a, b, c, and d.

### Expanding Both Sides

Let's begin by expanding both sides of the equation using the distributive property (also known as FOIL):

**Left Side:**
**(a + b + c + d)(a - b - c + d)** = a(a - b - c + d) + b(a - b - c + d) + c(a - b - c + d) + d(a - b - c + d)
= a² - ab - ac + ad + ab - b² - bc + bd + ac - bc - c² + cd + ad - bd - cd + d²
= **a² - b² - c² + d² + 2ad - 2bc**

**Right Side:**
**(a - b + c - d)(a + b - c - d)** = a(a + b - c - d) - b(a + b - c - d) + c(a + b - c - d) - d(a + b - c - d)
= a² + ab - ac - ad - ab - b² + bc + bd + ac + bc - c² - cd - ad - bd + cd + d²
= **a² - b² - c² + d² + 2ad - 2bc**

### Conclusion

As we can see, both the left and right sides of the equation simplify to the same expression: **a² - b² - c² + d² + 2ad - 2bc**. Therefore, the equation is true for any values of a, b, c, and d.

This demonstrates a key principle in algebra: expanding and simplifying expressions can help to reveal underlying relationships and prove equations.