1. **State the problem:** Simplify the expression $4x^{3} - 4x^{2} + 4x$. 2. **Identify the common factor:** Each term has a factor of $4x$. 3. **Factor out the common factor:** $$4x^{3} - 4x^{2} + 4x = 4x(x^{2} - x + 1)$$ 4. **Check if the quadratic can be factored further:** The quadratic $x^{2} - x + 1$ has discriminant $\Delta = (-1)^{2} - 4 \times 1 \times 1 = 1 - 4 = -3 < 0$, so it cannot be factored over the real numbers. 5. **Final simplified form:** $$4x(x^{2} - x + 1)$$ This is the fully factored form over the real numbers.