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.

Learn how to simplify the algebraic expression (a - b + c)(2a + b - c) by applying the distributive property. This step-by-step solution includes combining like terms and using key algebraic concepts. Suitable for students in Grades 8-10.

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