To differentiate the function \( y = -\frac{3x^2 - 6}{(x^2 - x - 2)^2} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative is given by:

\[
y' = \frac{u'v - uv'}{v^2}
\]

where \( u = -3x^2 + 6 \) and \( v = (x^2 - x - 2)^2 \).

1. **Differentiate \( u \)**:
   \[
   u' = \frac{d}{dx}(-3x^2 + 6) = -6x
   \]

2. **Differentiate \( v \)**:
   We first find the derivative of \( v = (x^2 - x - 2)^2 \) using the chain rule:
   \[
   v' = 2(x^2 - x - 2)(2x - 1)
   \]
   Here, \( \frac{d}{dx}(x^2 - x - 2) = 2x - 1 \).

3. **Apply the quotient rule**:
   Now we can substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule:
   \[
   y' = \frac{(-6x)(x^2 - x - 2)^2 - (-3x^2 + 6)(2(x^2 - x - 2)(2x - 1))}{(x^2 - x - 2)^4}
   \]

4. **Simplify**:
   This derivative can be simplified, but let's leave it in this form for clarity.

Thus, the derivative is:

\[
y' = \frac{-6x(x^2 - x - 2)^2 + (3x^2 - 6)(2(x^2 - x - 2)(2x - 1))}{(x^2 - x - 2)^4}
\]

If you need further simplification or have questions, let me know!

### Follow-up Questions:
1. Would you like a detailed step-by-step simplification of the derivative?
2. Are you interested in finding critical points of the function?
3. Would you like to know about the behavior of the function as \( x \) approaches certain values?
4. Do you want to graph the original function and its derivative?
5. Would you like to explore the applications of this derivative in real-world scenarios?

**Tip:** When using the quotient rule, always clearly identify \( u \) and \( v \) to avoid confusion during differentiation.

Explore the differentiation of the function -3x^2-6/(x^2-x-2)^2 using the quotient rule. This detailed solution covers key calculus concepts, including the application of the chain rule and simplification of derivatives. Ideal for high school students in Grades 11-12.

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