The function you've provided is:

\[
f(x) = \frac{x^3 - 3}{2x^2 + 4x^3}
\]

### Simplifying the expression:
We can factor both the numerator and the denominator to see if the expression can be simplified.

#### Step 1: Factor the numerator and denominator
- Numerator: \( x^3 - 3 \) (this is already factored as much as possible).
- Denominator: \( 2x^2 + 4x^3 \) can be factored as:
  \[
  2x^2(1 + 2x)
  \]

Thus, the function becomes:
\[
f(x) = \frac{x^3 - 3}{2x^2(1 + 2x)}
\]

#### Step 2: Analyze the simplified form
We cannot simplify this any further because there are no common factors between the numerator and the denominator. Therefore, the expression remains:

\[
f(x) = \frac{x^3 - 3}{2x^2(1 + 2x)}
\]

### Domain of the function:
The denominator \( 2x^2(1 + 2x) \) must not be equal to zero, which implies two conditions:
1. \( x^2 \neq 0 \) → \( x \neq 0 \)
2. \( 1 + 2x \neq 0 \) → \( x \neq -\frac{1}{2} \)

Thus, the domain of the function is all real numbers except \( x = 0 \) and \( x = -\frac{1}{2} \).

Would you like further details or clarifications on any part?

### Related questions:
1. How do you find critical points of a rational function?
2. What happens when a function has vertical asymptotes?
3. Can you find the derivative of the given function \( f(x) \)?
4. How can we analyze the behavior of \( f(x) \) near its vertical asymptotes?
5. How do we perform partial fraction decomposition on rational functions?

**Tip:** When analyzing rational functions, always check the domain by ensuring the denominator is never zero.

Explore the simplification of the rational function f(x) = (x^3 - 3) / (2x^2 + 4x^3) with step-by-step factoring and domain analysis. This problem introduces key algebraic concepts, such as factoring and determining the domain, and is suitable for Grades 9-12.

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