To determine the domain of the function

$f(x) = \frac{x^2 + 8}{5x^3 - 60x^2 + 12x + 99}$

we need to identify any values of $x$ that would make the denominator equal to zero, as the function would be undefined at those points.

Step 1: Solve for when the denominator is zero

The denominator is:

$5x^3 - 60x^2 + 12x + 99$

We need to solve the equation:

$5x^3 - 60x^2 + 12x + 99 = 0$

Step 2: Analyze the roots of the cubic equation

This is a cubic equation, and finding the roots can be done through techniques such as factoring (if possible), synthetic division, or using the Rational Root Theorem. In more advanced cases, numerical methods may be required.

For now, we will assume the roots of this cubic equation can be found. Let's denote these roots as $x_1, x_2, x_3$. These values of $x$ are excluded from the domain because they make the denominator zero.

Step 3: State the domain

The domain of $f(x)$ is all real numbers except for the values of $x$ that make the denominator zero. This can be expressed as:

$\text{Domain of } f(x) = \mathbb{R} \setminus \{x_1, x_2, x_3\}$

Would you like me to help find the exact values of $x_1, x_2, x_3$, or explain the next steps in more detail?

Related questions:

1. How do you factor a cubic equation?
2. What is the Rational Root Theorem?
3. What methods can be used to solve cubic equations?
4. How do you determine the asymptotes of a rational function?
5. What is the difference between the domain and range of a function?

Tip: When solving cubic equations, it's often helpful to check for simple integer roots using the Rational Root Theorem before applying more advanced methods.