Let's solve the given expression step by step.

We are given:

$-7(5a - b) + 3(-8b + a)$

Step 1: Distribute the numbers outside the parentheses.

• First, distribute $-7$ to both $5a$ and $-b$: $-7 \times 5a = -35a, \quad -7 \times (-b) = 7b$ Thus, the first part becomes: $-35a + 7b$

• Next, distribute $3$ to both $-8b$ and $a$: $3 \times (-8b) = -24b, \quad 3 \times a = 3a$ Thus, the second part becomes: $-24b + 3a$

Step 2: Combine like terms.

Now, combine the terms from both parts: $(-35a + 7b) + (-24b + 3a) = -35a + 3a + 7b - 24b$ Simplify: $-35a + 3a = -32a, \quad 7b - 24b = -17b$

Final expression:

$-32a - 17b$

This matches the last option in the list: $-32a - 17b$.

Answer:

The correct option is $-32a - 17b$.

Would you like further details or explanations? Here are five additional questions based on this problem:

1. How do you handle distribution in more complex expressions with multiple variables?
2. What happens when there are more than two sets of parentheses in an expression?
3. How can combining like terms help simplify polynomial expressions?
4. What role does the order of operations play in simplifying expressions like this one?
5. How do you check if your simplified expression is equivalent to one of the answer choices?

Tip: Always perform distribution carefully, especially when dealing with negative signs!