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.