The expression \((a + b + c)^3\) can be expanded using the binomial theorem or by manually expanding it. Here's how it works:

\[
(a + b + c)^3 = (a + b + c)(a + b + c)(a + b + c)
\]

Let's expand it step by step:

1. First, expand \((a + b + c)(a + b + c)\):

\[
(a + b + c)(a + b + c) = a^2 + ab + ac + ab + b^2 + bc + ac + bc + c^2
\]
\[
= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
\]

2. Now, multiply the result by \((a + b + c)\):

\[
(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc)(a + b + c)
\]

Expanding each term:

\[
= a^3 + a^2b + a^2c + ab^2 + abc + abc + ac^2 + b^3 + b^2a + b^2c + ba^2 + bac + bc^2 + c^3 + c^2a + c^2b + ca^2 + cab + cb^2
\]

Simplifying, we get:

\[
= a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3b^2c + 3ac^2 + 3bc^2 + 6abc
\]

So, the expanded form of \((a + b + c)^3\) is:

\[
\boxed{a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3ab^2 + 3b^2c + 3ac^2 + 3bc^2 + 6abc}
\]

Would you like to dive deeper into any part of this process, or do you have any questions about this expansion?

Here are five related questions you might find interesting:
1. How would you expand \((a + b)^4\)?
2. Can you generalize the expansion for \((a + b + c)^n\)?
3. What is the expanded form of \((2a + 3b + 4c)^3\)?
4. How does the expansion change if one of the variables is set to zero?
5. How can the binomial theorem be applied to expand \((a + b + c + d)^3\)?

**Tip:** When expanding binomials or trinomials, it's often helpful to group like terms to avoid errors and simplify the process.

Learn how to expand the expression (a + b + c)^3 using the binomial theorem with a step-by-step explanation. This problem covers polynomial expansion and the application of the binomial theorem. Suitable for students in Grades 10-12.

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