1. **State the problem:** Simplify the expression $$\frac{1}{2x - 6} + \frac{x^2 + 4x + 3}{x^2 + 2x - 15} + \frac{1}{x + 5}$$. 2. **Factor all denominators and numerators where possible:** - Factor $$2x - 6 = 2(x - 3)$$. - Factor $$x^2 + 2x - 15 = (x + 5)(x - 3)$$. - Factor numerator $$x^2 + 4x + 3 = (x + 3)(x + 1)$$. 3. **Rewrite the expression with factored forms:** $$\frac{1}{2(x - 3)} + \frac{(x + 3)(x + 1)}{(x + 5)(x - 3)} + \frac{1}{x + 5}$$ 4. **Find the least common denominator (LCD):** The denominators are $$2(x - 3)$$, $$(x + 5)(x - 3)$$, and $$(x + 5)$$. The LCD must include all factors: $$2(x - 3)(x + 5)$$. 5. **Rewrite each fraction with the LCD as denominator:** - First fraction: multiply numerator and denominator by $$(x + 5)$$: $$\frac{1 \cdot (x + 5)}{2(x - 3)(x + 5)} = \frac{x + 5}{2(x - 3)(x + 5)}$$ - Second fraction: multiply numerator and denominator by $$2$$: $$\frac{2(x + 3)(x + 1)}{2(x + 5)(x - 3)} = \frac{2(x + 3)(x + 1)}{2(x - 3)(x + 5)}$$ - Third fraction: multiply numerator and denominator by $$2(x - 3)$$: $$\frac{1 \cdot 2(x - 3)}{2(x - 3)(x + 5)} = \frac{2(x - 3)}{2(x - 3)(x + 5)}$$ 6. **Combine the fractions:** $$\frac{x + 5}{2(x - 3)(x + 5)} + \frac{2(x + 3)(x + 1)}{2(x - 3)(x + 5)} + \frac{2(x - 3)}{2(x - 3)(x + 5)} = \frac{(x + 5) + 2(x + 3)(x + 1) + 2(x - 3)}{2(x - 3)(x + 5)}$$ 7. **Expand and simplify the numerator:** - Expand $$2(x + 3)(x + 1)$$: $$2(x^2 + 4x + 3) = 2x^2 + 8x + 6$$ - Expand $$2(x - 3) = 2x - 6$$ - Sum all terms: $$x + 5 + 2x^2 + 8x + 6 + 2x - 6 = 2x^2 + (x + 8x + 2x) + (5 + 6 - 6) = 2x^2 + 11x + 5$$ 8. **Final simplified expression:** $$\frac{2x^2 + 11x + 5}{2(x - 3)(x + 5)}$$ 9. **Check if numerator can be factored:** Factor $$2x^2 + 11x + 5$$: - Factors of $$2 \times 5 = 10$$ that sum to 11 are 10 and 1. - Rewrite numerator: $$2x^2 + 10x + x + 5 = 2x(x + 5) + 1(x + 5) = (2x + 1)(x + 5)$$ 10. **Cancel common factor $$(x + 5)$$:** $$\frac{\cancel{(x + 5)}(2x + 1)}{2(x - 3)\cancel{(x + 5)}} = \frac{2x + 1}{2(x - 3)}$$ **Final answer:** $$\boxed{\frac{2x + 1}{2(x - 3)}}$$