Solving (a + b + c)(a + b  c)
This expression is a product of two binomials, and we can solve it using the distributive property or by recognizing a pattern.
Using the Distributive Property
The distributive property states that a(b + c) = ab + ac. We can apply this to our expression:

Expand the first binomial: (a + b + c)(a + b  c) = a(a + b  c) + b(a + b  c) + c(a + b  c)

Distribute: = a² + ab  ac + ab + b²  bc + ac + bc  c²

Combine like terms: = a² + 2ab + b²  c²
Recognizing a Pattern
The expression (a + b + c)(a + b  c) resembles the difference of squares pattern: (x + y)(x  y) = x²  y².
In our case, we can consider (a + b) as 'x' and 'c' as 'y'.
Therefore:
(a + b + c)(a + b  c) = (a + b)²  c²
This can be further expanded as:
= a² + 2ab + b²  c²
Conclusion
Both methods lead to the same answer: a² + 2ab + b²  c².
This expression represents the simplified form of the product (a + b + c)(a + b  c).