Expanding (a + b + c)(a + b  c)
This expression represents the product of two binomials. To expand it, we can use the distributive property or the FOIL method.
Using the Distributive Property:

Treat (a + b + c) as a single term and distribute it to the second binomial: (a + b + c)(a + b  c) = (a + b + c) * a + (a + b + c) * b + (a + b + c) * (c)

Distribute again within each of the resulting terms: = a² + ab + ac + ab + b² + bc  ac  bc  c²

Combine like terms: = a² + 2ab + b²  c²
Using the FOIL Method:
The FOIL method is a mnemonic acronym for First, Outer, Inner, Last. It helps us remember which terms to multiply together.

Multiply the First terms: a * a = a²

Multiply the Outer terms: a * (c) = ac

Multiply the Inner terms: b * a = ab

Multiply the Last terms: b * (c) = bc

Multiply the First term of the first binomial by the remaining terms in the second binomial: (a + b + c) * b = ab + b² + bc

Multiply the second term of the first binomial by the remaining terms in the second binomial: (a + b + c) * (c) = ac  bc  c²

Combine all the terms: = a²  ac + ab  bc + ab + b² + bc  ac  c²

Combine like terms: = a² + 2ab + b²  c²
Therefore, the expanded form of (a + b + c)(a + b  c) is a² + 2ab + b²  c².
Important Note: This expression can also be factored as a difference of squares: (a + b + c)(a + b  c) = [(a + b) + c][(a + b)  c] = (a + b)²  c². This can be helpful in solving equations or simplifying expressions.