The problem asks to evaluate the composition of functions $(g \circ f)(x)$ for $x = 3$, where:

$f(x) = x^3 + 8x, \quad g(x) = \sqrt{5x}, \quad h(x) = 3x + 4$

Step-by-step solution:

1. First, compute $f(3)$:

$f(3) = 3^3 + 8 \times 3 = 27 + 24 = 51$

2. Next, compute $g(f(3))$, which is $g(51)$:

$g(51) = \sqrt{5 \times 51} = \sqrt{255}$

Since $\sqrt{255}$ cannot be simplified further, the final answer is:

$(g \circ f)(3) = \sqrt{255}$

Let me know if you need more details or have questions.

Here are 5 related questions to explore:

1. How do you find the composition of functions in general?
2. How does the domain of a composition of functions affect its evaluation?
3. Can you simplify the square root of 255 any further?
4. How would the result change if $f(x)$ was a different polynomial?
5. What is the significance of function composition in real-world problems?

Tip: When solving composition of functions, always calculate from the innermost function outward!