Question 1.2 \( 2 \div\left(\frac{a+b}{a+2 b} \times \frac{b^{2}-b a-6 a^{2}}{a^{2}-4 b^{2}}\right) \)

Let's tackle this step by step! To simplify the expression, start by focusing on the fraction inside the parentheses. The expression can benefit from factoring, especially in the second fraction. First, note that the quadratic \(b^{2} - ba - 6a^{2}\) can be factored. The factors of \(-6a^{2}\) that add up to \(-b\) are \(-3a\) and \(2a\), leading to \((b - 3a)(b + 2a)\). The denominator \(a^{2} - 4b^{2}\) is a difference of squares, factoring to \((a - 2b)(a + 2b)\). Once simplified, plug in any values for \(a\) and \(b\) to see how changes affect the entire expression and practice ensures you'll master this! Additionally, always double-check your factorizations and sign changes. It's easy to misplace a negative sign or misinterpret the multiplicative factors, leading you down a slippery slope of mistakes. Staying organized will save you time and frustration!