simplify: 2[9(A-B) + 7(2B-A)] - 2[3(2B+A) - 2(A+3B) - 5(A+B)]

Let's simplify the given expression step by step:

The expression is:

\[
2[9(A-B) + 7(2B - A)] - 2[3(2B + A) - 2(A + 3B) - 5(A + B)]
\]

### Step 1: Simplify inside the first bracket
Expand the terms inside the first set of brackets:
\[
9(A - B) = 9A - 9B
\]
\[
7(2B - A) = 14B - 7A
\]
Now, combine these:
\[
9A - 9B + 14B - 7A = (9A - 7A) + (-9B + 14B) = 2A + 5B
\]
Thus, the first bracket becomes:
\[
2[2A + 5B]
\]

### Step 2: Simplify inside the second bracket
Expand the terms inside the second set of brackets:
\[
3(2B + A) = 6B + 3A
\]
\[
-2(A + 3B) = -2A - 6B
\]
\[
-5(A + B) = -5A - 5B
\]
Now, combine these:
\[
(6B + 3A) + (-2A - 6B) + (-5A - 5B) = (3A - 2A - 5A) + (6B - 6B - 5B) = -4A - 5B
\]
Thus, the second bracket becomes:
\[
2[-4A - 5B]
\]

### Step 3: Apply the factors outside the brackets
Now multiply the factors outside the brackets:
\[
2[2A + 5B] = 4A + 10B
\]
\[
2[-4A - 5B] = -8A - 10B
\]

### Step 4: Combine the results
Now combine the two results:
\[
(4A + 10B) - (-8A - 10B) = 4A + 10B + 8A + 10B = (4A + 8A) + (10B + 10B) = 12A + 20B
\]

Thus, the simplified expression is:

\[
\boxed{12A + 20B}
\]

---

Would you like further clarification on any part of this solution?

Here are five related questions to extend your understanding:
1. How would you simplify a similar expression with three variables?
2. What is the distributive property, and how does it apply in this case?
3. Can this result be factored? If so, how?
4. What is the general approach to simplify expressions involving nested brackets?
5. How would the expression change if the sign of one of the terms inside the brackets were different?

**Tip:** Always simplify inside the brackets first, then apply the multiplication outside the brackets step by step.

Simplify the algebraic expression 2[9(A-B) + 7(2B-A)] - 2[3(2B+A) - 2(A+3B) - 5(A+B)] step by step. This problem involves applying the distributive property and combining like terms, perfect for students in Grades 8-10.

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