## Factoring the Expression (a+b+c)(a+b-c)(b+c-a)(c+a-b)

This expression appears complex at first glance, but it can be simplified significantly using a clever factorization technique. Let's explore how to break it down:

### Understanding the Pattern

Notice that the terms within the parentheses follow a specific pattern:

**First factor:**(a + b + c)**Second factor:**(a + b - c)**Third factor:**(b + c - a)**Fourth factor:**(c + a - b)

Each factor has two positive terms and one negative term, and the negative term changes its sign systematically. This pattern is crucial for the factorization.

### Applying the Difference of Squares

Let's group the factors strategically:

**Group 1:**(a + b + c)(a + b - c)**Group 2:**(b + c - a)(c + a - b)

Now, we can apply the difference of squares pattern:

**(x + y)(x - y) = x² - y²**

Applying this to each group:

**Group 1:**(a + b)² - c²**Group 2:**(b + c)² - a²

### Expanding and Simplifying

Let's expand the squares and rearrange the terms:

**Group 1:**a² + 2ab + b² - c²**Group 2:**b² + 2bc + c² - a²

Now, we have:

**(a² + 2ab + b² - c²)(b² + 2bc + c² - a²) **

Observe that the terms involving a² and b² cancel out:

**(2ab + 2bc + 2ac)(2ab + 2bc + 2ac)**

Finally, we can factor out a common factor of 2:

**(2)(ab + bc + ac)(2)(ab + bc + ac)**

**Therefore, the simplified expression is:** 4(ab + bc + ac)²

### Conclusion

By recognizing the pattern in the expression and applying the difference of squares, we successfully factored the expression into a much simpler form: 4(ab + bc + ac)². This method highlights the importance of identifying patterns and applying algebraic identities to simplify complex expressions.