Let's simplify the given expression step-by-step:

Expression:

$\frac{4x^2(3 - x)}{6x(3 - x)}$

Step 1: Cancel common terms

The factor $(3 - x)$ appears in both the numerator and denominator, so it cancels out (as long as $3 - x \neq 0$, i.e., $x \neq 3$):

$\frac{4x^2}{6x}$

Step 2: Simplify the remaining fraction

Now simplify the remaining terms:

$\frac{4x^2}{6x} = \frac{4}{6} \cdot \frac{x^2}{x} = \frac{2}{3} \cdot x = \frac{2x}{3}$

Final Answer:

$\frac{2x}{3}, \quad \text{for } x \neq 0 \text{ and } x \neq 3.$

If you have more conditions or want further clarification, let me know!

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Would you like additional details on this problem? Here are five related questions:

1. What happens if $x = 0$ or $x = 3$?
2. Can we analyze this problem graphically or in terms of functions?
3. How do cancellations in algebra work conceptually?
4. What if we expand $3 - x$ instead of factoring?
5. How do restrictions on $x$ affect the domain of this expression?

Tip: Always state restrictions on variables clearly when simplifying rational expressions!