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The given expressions are:

\[
A = 4a^3 + 3a - 19
\]
\[
B = a^3 + 8a^2 - 7a + 13
\]
\[
C = -5a^3 - 6a + 20
\]

You are asked to find \( A + B - C \). Let's calculate it step by step.

### Step 1: Add \( A \) and \( B \)
\[
A + B = (4a^3 + 3a - 19) + (a^3 + 8a^2 - 7a + 13)
\]
Simplifying:
\[
A + B = (4a^3 + a^3) + 8a^2 + (3a - 7a) + (-19 + 13)
\]
\[
A + B = 5a^3 + 8a^2 - 4a - 6
\]

### Step 2: Subtract \( C \) from \( A + B \)
\[
A + B - C = (5a^3 + 8a^2 - 4a - 6) - (-5a^3 - 6a + 20)
\]
Simplifying:
\[
A + B - C = 5a^3 + 8a^2 - 4a - 6 + 5a^3 + 6a - 20
\]
\[
A + B - C = (5a^3 + 5a^3) + 8a^2 + (-4a + 6a) + (-6 - 20)
\]
\[
A + B - C = 10a^3 + 8a^2 + 2a - 26
\]

### Final Answer:
\[
A + B - C = 10a^3 + 8a^2 + 2a - 26
\]

Would you like further clarification or details on any step? 

Here are some related questions for further exploration:
1. How would the result change if the coefficients of \(a^2\) in \(A\) were different?
2. Can you verify the result by substituting a specific value of \(a\)?
3. How do the operations of polynomial addition and subtraction differ when dealing with higher-degree terms?
4. What happens if the expression includes a term like \( a^4 \)?
5. How can you simplify complex expressions like this efficiently?

**Tip:** When adding or subtracting polynomials, group similar terms (like powers of \(a\)) to make simplification easier.

If A = 4a^3 + 3a - 19, B = a^3 + 8a^2 - 7a + 13 and C = -5a^3 - 6a + 20, find A + B - C.

Solve the algebraic problem involving the polynomials A = 4a^3 + 3a - 19, B = a^3 + 8a^2 - 7a + 13, and C = -5a^3 - 6a + 20. Learn how to simplify A + B - C by combining like terms and performing polynomial operations. Suitable for Grades 9-10.

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