To simplify the expression $(a - b + c)(2a + b - c)$, we will apply the distributive property (also known as FOIL in binomials):

$(a - b + c)(2a + b - c)$

First, distribute each term from the first expression to each term in the second:

1. $a \times 2a = 2a^2$
2. $a \times b = ab$
3. $a \times (-c) = -ac$
4. $-b \times 2a = -2ab$
5. $-b \times b = -b^2$
6. $-b \times (-c) = bc$
7. $c \times 2a = 2ac$
8. $c \times b = bc$
9. $c \times (-c) = -c^2$

Now, combine all these terms:

$2a^2 + ab - ac - 2ab - b^2 + bc + 2ac + bc - c^2$

Next, group and combine like terms:

$2a^2 + (ab - 2ab) + (-ac + 2ac) + (-b^2) + (bc + bc) - c^2$

Simplifying:

$2a^2 - ab + ac - b^2 + 2bc - c^2$

Thus, the simplified expression is:

$\boxed{2a^2 - ab + ac - b^2 + 2bc - c^2}$

Would you like further details on this? Here are some follow-up questions that might help you explore this concept further:

1. How do we apply the distributive property to more complex expressions?
2. Can you factor the resulting expression further if given specific values for $a$, $b$, and $c$?
3. What are the differences between factoring and expanding expressions?
4. How do like terms help in simplifying algebraic expressions?
5. How does this approach change if we were simplifying a trinomial instead?

Tip: Always watch out for common terms when simplifying algebraic expressions, as they can often be combined to make the process easier.