1. **State the problem:** Simplify the expression $$\frac{x^2 + 4}{x^2 - 7x + 6} - \frac{8}{x - 6}$$. 2. **Factor the denominator:** The quadratic in the denominator factors as $$x^2 - 7x + 6 = (x - 6)(x - 1)$$. 3. **Rewrite the expression:** $$\frac{x^2 + 4}{(x - 6)(x - 1)} - \frac{8}{x - 6}$$ 4. **Find a common denominator:** The common denominator is $(x - 6)(x - 1)$. 5. **Rewrite the second fraction with the common denominator:** $$\frac{8}{x - 6} = \frac{8(x - 1)}{(x - 6)(x - 1)}$$ 6. **Combine the fractions:** $$\frac{x^2 + 4}{(x - 6)(x - 1)} - \frac{8(x - 1)}{(x - 6)(x - 1)} = \frac{x^2 + 4 - 8(x - 1)}{(x - 6)(x - 1)}$$ 7. **Simplify the numerator:** $$x^2 + 4 - 8(x - 1) = x^2 + 4 - 8x + 8 = x^2 - 8x + 12$$ 8. **Factor the numerator:** $$x^2 - 8x + 12 = (x - 6)(x - 2)$$ 9. **Rewrite the expression:** $$\frac{(x - 6)(x - 2)}{(x - 6)(x - 1)}$$ 10. **Cancel the common factor $(x - 6)$:** $$\frac{\cancel{(x - 6)}(x - 2)}{\cancel{(x - 6)}(x - 1)} = \frac{x - 2}{x - 1}$$ **Final answer:** $$\boxed{\frac{x - 2}{x - 1}}$$ **Note:** The domain excludes $x = 6$ and $x = 1$ because they make the original denominator zero.