3a-3b/5a+10b•a+2b/a-2b•(10a-20b)

To simplify the expression \( \frac{3a - 3b}{5a + 10b} \cdot \frac{a + 2b}{a - 2b} \cdot (10a - 20b) \), let's proceed step by step.

1. Factor the components of the expression:

• The first fraction: \[ \frac{3a - 3b}{5a + 10b} = \frac{3(a - b)}{5(a + 2b)} \]

• The second fraction: \[ \frac{a + 2b}{a - 2b} \quad \text{(no factoring needed here)} \]

• The third term: \[ 10a - 20b = 10(a - 2b) \]

2. Substitute the factored terms back into the expression: \[ \frac{3(a - b)}{5(a + 2b)} \cdot \frac{a + 2b}{a - 2b} \cdot 10(a - 2b) \]

3. Combine everything: \[ = \frac{3(a - b) \cdot 10(a - 2b)}{5(a + 2b)(a - 2b)} \]

4. Simplify by canceling \( (a - 2b) \) from the numerator and the denominator: \[ = \frac{3 \cdot 10(a - b)}{5(a + 2b)} = \frac{30(a - b)}{5(a + 2b)} \]

5. Further simplify \( \frac{30}{5} = 6 \): \[ = \frac{6(a - b)}{a + 2b} \]

Thus, the simplified form of the original expression is: \[ \frac{6(a - b)}{a + 2b} \]