1. **State the problem:** We want to evaluate the integral $$\int \frac{2x - 8}{x^2 - 8x + 32} \, dx.$$\n\n2. **Rewrite the denominator:** Notice that $$x^2 - 8x + 32 = (x - 4)^2 + 16.$$ This is a sum of a square and a positive constant, which suggests using substitution or recognizing derivative patterns.\n\n3. **Check the numerator:** The numerator is $$2x - 8 = 2(x - 4).$$ This is proportional to the derivative of the denominator's inner function $(x-4)^2$.\n\n4. **Use substitution:** Let $$u = x^2 - 8x + 32 = (x - 4)^2 + 16.$$ Then, $$\frac{du}{dx} = 2x - 8,$$ which matches the numerator exactly.\n\n5. **Rewrite the integral:** Using substitution, $$\int \frac{2x - 8}{x^2 - 8x + 32} \, dx = \int \frac{du}{u}.$$\n\n6. **Integrate:** $$\int \frac{du}{u} = \ln|u| + C = \ln|x^2 - 8x + 32| + C.$$\n\n7. **Final answer:** $$\boxed{\ln\left((x - 4)^2 + 16\right) + C}.$$\n\nThis matches the third option in the problem's choices and is the correct integral result.