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.