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.