Step 1 :First, we find the composition of functions $f(g(x))$.
Step 2 :$$f(g(x)) = f((x-2)^2) = (x-2)^2 + 4$$
Step 3 :Now, we want to find a function $u(x)$ such that $f(g(u(x))) = 4x^2 - 8x + 8$.
Step 4 :We can rewrite the given function as:
Step 5 :$$4x^2 - 8x + 8 = (2x-4)^2$$
Step 6 :Comparing this with the expression for $f(g(x))$, we can see that $u(x) = 2x-4$.
Step 7 :So, the function $u(x)$ is:
Step 8 :$$\boxed{u(x) = 2x-4}$$