To find the horizontal asymptote of the function:

$f(x) = \frac{-4x + 3x^3 + 5}{2x^3 + 4x^2 - 2}$

We need to consider the degrees of the numerator and the denominator.

Step-by-step process:

1. Degree of the numerator (top): The highest power of $x$ in the numerator is $3x^3$, so the degree of the numerator is 3.

2. Degree of the denominator (bottom): The highest power of $x$ in the denominator is $2x^3$, so the degree of the denominator is also 3.

3. Horizontal asymptote rules:

   • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
   • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
   • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.

   Here, both degrees are 3, so we take the leading coefficients:

   • Leading coefficient of the numerator is $3$.
   • Leading coefficient of the denominator is $2$.

   Therefore, the horizontal asymptote is:

   $y = \frac{3}{2}$

Final Answer:

$y = \frac{3}{2}$

Would you like a further breakdown or explanation?

Here are 5 related questions:

1. What happens if the degree of the numerator is one more than the degree of the denominator?
2. Can a rational function have more than one horizontal asymptote?
3. How do vertical asymptotes differ from horizontal asymptotes?
4. What is the end behavior of a function with no horizontal asymptote?
5. How can slant asymptotes occur in rational functions?

Tip: The horizontal asymptote describes the behavior of the function as $x$ approaches infinity or negative infinity.