1. **State the problem:** Simplify the expression $$\frac{k^2 - 17k + 39}{k^2 - 3k - 10} + \frac{3}{k - 5}$$. 2. **Factor the polynomials:** - Factor the numerator of the first fraction: $$k^2 - 17k + 39$$. - Factor the denominator of the first fraction: $$k^2 - 3k - 10$$. 3. **Factoring the numerator:** We look for two numbers that multiply to 39 and add to -17. These are -13 and -3. $$k^2 - 17k + 39 = (k - 13)(k - 3)$$ 4. **Factoring the denominator:** We look for two numbers that multiply to -10 and add to -3. These are -5 and 2. $$k^2 - 3k - 10 = (k - 5)(k + 2)$$ 5. **Rewrite the expression with factored forms:** $$\frac{(k - 13)(k - 3)}{(k - 5)(k + 2)} + \frac{3}{k - 5}$$ 6. **Find common denominator:** The common denominator is $$(k - 5)(k + 2)$$. Rewrite the second fraction: $$\frac{3}{k - 5} = \frac{3(k + 2)}{(k - 5)(k + 2)}$$ 7. **Combine the fractions:** $$\frac{(k - 13)(k - 3)}{(k - 5)(k + 2)} + \frac{3(k + 2)}{(k - 5)(k + 2)} = \frac{(k - 13)(k - 3) + 3(k + 2)}{(k - 5)(k + 2)}$$ 8. **Expand the numerator:** $$(k - 13)(k - 3) = k^2 - 3k - 13k + 39 = k^2 - 16k + 39$$ $$3(k + 2) = 3k + 6$$ 9. **Add the expanded terms:** $$k^2 - 16k + 39 + 3k + 6 = k^2 - 13k + 45$$ 10. **Rewrite the expression:** $$\frac{k^2 - 13k + 45}{(k - 5)(k + 2)}$$ 11. **Factor the numerator if possible:** Look for two numbers that multiply to 45 and add to -13. These are -9 and -5. $$k^2 - 13k + 45 = (k - 9)(k - 5)$$ 12. **Simplify the fraction by canceling common factors:** $$\frac{(k - 9)(k - 5)}{(k - 5)(k + 2)} = \frac{\cancel{(k - 5)}(k - 9)}{\cancel{(k - 5)}(k + 2)} = \frac{k - 9}{k + 2}$$ **Final answer:** $$\boxed{\frac{k - 9}{k + 2}}$$