Refer to question 5. Factor $f(x)$ over the real numbers. \[ f(x)=x^{3}-8 x^{2}+16 x-8 \] \[ f(x)= \]

Step 1 ：Given the cubic function \(f(x) = x^{3} - 8x^{2} + 16x - 8\).

Step 2 ：To factorize it, we first find the roots of the function. The roots of the function are the values of x for which \(f(x) = 0\).

Step 3 ：By solving the equation \(f(x) = 0\), we find the roots to be 2, \(3 - \sqrt{5}\), and \(\sqrt{5} + 3\).

Step 4 ：Once we find the roots, we can express the function as a product of \((x - \text{root})\) terms.

Step 5 ：Substituting the roots into the equation, we get the factored form of the function \(f(x) = (x - 2)(x - 3 - \sqrt{5})(x - 3 + \sqrt{5})\).

Step 6 ：\(\boxed{f(x) = (x - 2)(x - 3 - \sqrt{5})(x - 3 + \sqrt{5})}\) is the final answer.