1. **State the problem:** Simplify the expression $$\frac{2x - 6 + (x^2 + 4x + 3)}{x^2 + 2x - 15} + \frac{1}{x + 5}$$ 2. **Rewrite the expression clearly:** $$\frac{2x - 6 + x^2 + 4x + 3}{x^2 + 2x - 15} + \frac{1}{x + 5}$$ 3. **Simplify the numerator of the first fraction:** $$2x - 6 + x^2 + 4x + 3 = x^2 + (2x + 4x) + (-6 + 3) = x^2 + 6x - 3$$ 4. **Factor the denominator of the first fraction:** $$x^2 + 2x - 15 = (x + 5)(x - 3)$$ 5. **Rewrite the expression with simplified numerator and factored denominator:** $$\frac{x^2 + 6x - 3}{(x + 5)(x - 3)} + \frac{1}{x + 5}$$ 6. **Find a common denominator:** The common denominator is $(x + 5)(x - 3)$. 7. **Rewrite the second fraction with the common denominator:** $$\frac{1}{x + 5} = \frac{1 \cdot (x - 3)}{(x + 5)(x - 3)} = \frac{x - 3}{(x + 5)(x - 3)}$$ 8. **Add the two fractions:** $$\frac{x^2 + 6x - 3}{(x + 5)(x - 3)} + \frac{x - 3}{(x + 5)(x - 3)} = \frac{x^2 + 6x - 3 + x - 3}{(x + 5)(x - 3)} = \frac{x^2 + 7x - 6}{(x + 5)(x - 3)}$$ 9. **Factor the numerator if possible:** We look for factors of $-6$ that add to $7$. $$x^2 + 7x - 6 = (x + 6)(x - 1)$$ 10. **Rewrite the expression:** $$\frac{(x + 6)(x - 1)}{(x + 5)(x - 3)}$$ 11. **Final answer:** $$\boxed{\frac{(x + 6)(x - 1)}{(x + 5)(x - 3)}}$$ This is the simplified form of the original expression.