Let's break down the expression step by step:

Given expression:
$(6z^4 + 3z^2 - 9) \left( 3z^2 - 6 \right)^{-1}$

Step 1: Understand the Meaning of $(3z^2 - 6)^{-1}$

The term $(3z^2 - 6)^{-1}$ refers to the reciprocal (or inverse) of $3z^2 - 6$. In other words, $(3z^2 - 6)^{-1}$ is the same as: $\frac{1}{3z^2 - 6}$

Step 2: Rewrite the Expression

Now that we know what $(3z^2 - 6)^{-1}$ means, we can rewrite the original expression as: $\frac{6z^4 + 3z^2 - 9}{3z^2 - 6}$

Step 3: Simplify the Denominator

Notice that the denominator, $3z^2 - 6$, can be factored: $3z^2 - 6 = 3(z^2 - 2)$

So the expression becomes: $\frac{6z^4 + 3z^2 - 9}{3(z^2 - 2)}$

Step 4: Factor the Numerator

Next, let's factor the numerator, $6z^4 + 3z^2 - 9$. The first thing we can do is factor out a 3: $6z^4 + 3z^2 - 9 = 3(2z^4 + z^2 - 3)$

Now the expression looks like this: $\frac{3(2z^4 + z^2 - 3)}{3(z^2 - 2)}$

Step 5: Cancel Common Factors

Since both the numerator and denominator have a factor of 3, we can cancel that: $\frac{2z^4 + z^2 - 3}{z^2 - 2}$

Step 6: Final Expression

The simplified expression is: $\frac{2z^4 + z^2 - 3}{z^2 - 2}$

This is the simplified form of the given expression. It can't be simplified further unless you use specific values of $z$.

---

Do you want further details or explanations on any part of this?

Here are 5 related questions you might find useful:

1. How do I solve equations with rational expressions?
2. What are some strategies for factoring polynomials?
3. How do you handle negative exponents in more complex expressions?
4. How can I simplify a rational expression when there are variables in both the numerator and denominator?
5. How do you solve equations with complex fractions?

Tip: Always check for common factors between the numerator and denominator before simplifying a rational expression.