1. Problem: Factorize and simplify the expression $\cos 2x + \cos 4x + \cos 6x$. 2. Use the sum-to-product formulas: $\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$. 3. First, group $\cos 2x + \cos 6x$: $$\cos 2x + \cos 6x = 2 \cos \frac{2x + 6x}{2} \cos \frac{2x - 6x}{2} = 2 \cos 4x \cos (-2x) = 2 \cos 4x \cos 2x$$ 4. Now the expression becomes: $$2 \cos 4x \cos 2x + \cos 4x = \cos 4x (2 \cos 2x + 1)$$ 5. This is the factorized and simplified form: $$\boxed{\cos 4x (2 \cos 2x + 1)}$$