1. **State the problem:** Find the excluded values for the rational expression $$\frac{8x^2 - 5x + 7}{x^2 - 6x - 27}$$. 2. **Important rule:** Excluded values are the values of $x$ that make the denominator zero because division by zero is undefined. 3. **Set the denominator equal to zero:** $$x^2 - 6x - 27 = 0$$ 4. **Factor the quadratic:** We look for two numbers that multiply to $-27$ and add to $-6$. These are $-9$ and $3$. $$x^2 - 6x - 27 = (x - 9)(x + 3)$$ 5. **Set each factor equal to zero:** $$x - 9 = 0 \implies x = 9$$ $$x + 3 = 0 \implies x = -3$$ 6. **Conclusion:** The excluded values are $x = -3$ and $x = 9$. The smallest excluded value is $-3$. **Final answer:** $$x \neq -3, 9$$