The given function is:

$f(x) = (x + 4)^3 - 5$

To find the range, let's analyze the function step-by-step:

1. The expression $(x + 4)^3$ represents a cubic function, which has a domain of all real numbers ($\mathbb{R}$) and a range of all real numbers ($\mathbb{R}$) because cubic functions increase without bound in both directions.

2. Shifting this cubic function by 4 units to the left ($x + 4$) does not affect the range, so the range of $(x + 4)^3$ is still all real numbers.

3. The next step is the subtraction of 5. Subtracting a constant shifts the entire graph down by 5 units. Thus, the range of $f(x)$ is also all real numbers ($\mathbb{R}$).

Therefore, the range of the function $f(x) = (x + 4)^3 - 5$ is:

$\text{Range of } f(x) = (-\infty, \infty)$

Would you like more details or further explanations?

Here are 5 related questions:

1. What is the domain of a cubic function?
2. How do transformations affect the graph of a cubic function?
3. How would adding or subtracting a constant inside the cube affect the range?
4. Can a cubic function have a limited range?
5. What are the key points to consider when graphing a cubic function?

Tip: Cubic functions are always increasing or decreasing, which means they have no maximum or minimum points.