Given the equation \(a + b + c = 0\), we need to find the value of the expression \(a^2 + a^2c - abc + b^2c + b^3\).

First, let's simplify the given expression step-by-step.

### Simplify the Expression
1. Start with the expression:
   \[
   a^2 + a^2c - abc + b^2c + b^3
   \]

2. Combine like terms involving \(a\):
   \[
   a^2(1 + c) - abc + b^2c + b^3
   \]

3. Factor out common terms from \(a\) and \(b\):
   \[
   a^2(1 + c) - abc + b^2c + b^3
   \]

### Substitute \(c = -(a + b)\) into the Expression
Given \(a + b + c = 0\), we have:
\[
c = -(a + b)
\]

Substitute \(c = -(a + b)\) into the expression:
\[
a^2 + a^2(-a - b) - ab(-a - b) + b^2(-a - b) + b^3
\]

### Simplify Each Term
1. For \(a^2(1 + c)\):
   \[
   a^2(1 + (-a - b)) = a^2(1 - a - b) = a^2 - a^3 - a^2b
   \]

2. For \(-abc\):
   \[
   -ab(-a - b) = ab(a + b) = a^2b + ab^2
   \]

3. For \(b^2c\):
   \[
   b^2(-a - b) = -ab^2 - b^3
   \]

4. For \(b^3\):
   \[
   b^3 = b^3
   \]

### Combine the Simplified Terms
Combining all the simplified terms, we get:
\[
a^2 - a^3 - a^2b + a^2b + ab^2 - ab^2 - b^3 + b^3
\]

### Cancel Out Terms
Notice that \(a^2b - a^2b\), \(ab^2 - ab^2\), and \(-b^3 + b^3\) cancel out:
\[
a^2 - a^3
\]

So, the final simplified expression is:
\[
a^2 - a^3
\]

Therefore, given \(a + b + c = 0\), the value of \(a^2 + a^2c - abc + b^2c + b^3\) is:
\[
\boxed{a^2 - a^3}
\]

Would you like more details or have any questions?

### Related Questions
1. What happens to the expression if \(a + b + c = 1\) instead?
2. How does the expression change if one of the variables \(a, b,\) or \(c\) is zero?
3. Can you derive a general formula for expressions involving \(a, b, c\) with different exponents?
4. How does the expression simplify if \(a = b\)?
5. What if the expression includes a higher degree polynomial in terms of \(a\) and \(b\)?
6. Can we factorize the given expression further if \(a + b + c \neq 0\)?
7. How would the expression change if additional terms like \(a^3\) or \(b^4\) were included?
8. How does substituting \(a + b + c = 0\) help in simplifying more complex algebraic expressions?

### Tip
Always substitute given conditions early in the simplification process to reduce complexity and potentially cancel out terms.

Explore how to simplify the expression a^2 + a^2c - abc + b^2c + b^3 with a step-by-step solution. This problem involves algebraic manipulation, factorization, and substitution techniques. Suitable for advanced high school students and beyond.

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