Expanding and Simplifying (a + b + c)²  (a  b  c)²
This expression involves squaring binomials and then subtracting the results. We can simplify it using the following steps:
1. Expanding the Squares:
We can expand the squares using the algebraic identity: (x + y)² = x² + 2xy + y².

(a + b + c)²: Let's treat (b + c) as a single term.
(a + b + c)² = a² + 2a(b + c) + (b + c)²
Expanding further: a² + 2ab + 2ac + b² + 2bc + c²

(a  b  c)²: Similarly, treating (b  c) as a single term.
(a  b  c)² = a² + 2a(b  c) + (b  c)²
Expanding further: a²  2ab  2ac + b² + 2bc + c²
2. Subtracting the Expanded Expressions:
Now we substitute the expanded expressions back into the original expression:
(a + b + c)²  (a  b  c)² = (a² + 2ab + 2ac + b² + 2bc + c²)  (a²  2ab  2ac + b² + 2bc + c²)
3. Simplifying the Expression:
Notice that many terms cancel out:
 a² and a² cancel out.
 b² and b² cancel out.
 c² and c² cancel out.
This leaves us with:
2ab + 2ac + 2ab + 2ac = 4ab + 4ac
Therefore, (a + b + c)²  (a  b  c)² simplifies to 4ab + 4ac.
Key Takeaway: This problem illustrates the power of using algebraic identities to simplify complex expressions. By strategically applying identities, we can often arrive at a simpler and more manageable form.