To construct a rational function that satisfies the given conditions:

1. Vertical Asymptotes occur when the denominator equals zero. For vertical asymptotes at $x = -4$ and $x = 5$, the denominator must have the factors $(x + 4)$ and $(x - 5)$.

2. X-intercept occurs when the numerator equals zero. For an x-intercept at $(-2, 0)$, the numerator must have the factor $(x + 2)$.

Putting these together, the rational function can be written as:

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

Here, $a$ is a constant that can be any non-zero value. If no specific value of $a$ is given, we can set $a = 1$ for simplicity:

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

Let me know if you'd like further clarifications or details!

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Related Questions:

1. How can we determine horizontal asymptotes for this function?
2. What happens to the function's behavior near the vertical asymptotes?
3. What is the domain of this rational function?
4. How would the graph of this function look?
5. How do changes in the constant $a$ affect the function?

Tip: When writing rational functions, always double-check the placement of factors to ensure they satisfy the given conditions.