To solve the problem given, we first need to compute $g(f(8))$.

1. Calculate $f(8)$: $f(x) = x^2 - 8 \implies f(8) = 8^2 - 8 = 64 - 8 = 56$

2. Then compute $g(f(8))$ or $g(56)$: $g(x) = 4x + 12 \implies g(56) = 4 \times 56 + 12 = 224 + 12 = 236$

Therefore, $g(f(8)) = 236$. Now, we check if $g(f(8))$ exists, which it does, hence we will select option A and the answer is 236.

If you have any further questions or need more details on this solution, please let me know! Here are some related questions you might find interesting:

1. What happens if you swap the functions, calculating $f(g(8))$?
2. How would the answer change if $g(x)$ was defined differently, such as $g(x) = x + 4$?
3. Can you derive $g(f(x))$ for a general $x$ and not just for $x = 8$?
4. What is the significance of the composition of functions in algebra?
5. How can graphing $f(x)$ and $g(x)$ help in understanding their composition?

Tip: When dealing with function compositions, always compute the inner function first before applying the outer function to the result.