1. **State the problem:** Simplify the expression $$\frac{5}{(x-7)(x+5)} + \frac{2}{(x-7)(x+3)}$$ and express the answer in factored form. 2. **Find a common denominator:** The denominators are $(x-7)(x+5)$ and $(x-7)(x+3)$. The least common denominator (LCD) is $$(x-7)(x+5)(x+3)$$ because both denominators share the factor $(x-7)$. 3. **Rewrite each fraction with the LCD:** $$\frac{5}{(x-7)(x+5)} = \frac{5(x+3)}{(x-7)(x+5)(x+3)}$$ $$\frac{2}{(x-7)(x+3)} = \frac{2(x+5)}{(x-7)(x+5)(x+3)}$$ 4. **Add the numerators over the common denominator:** $$\frac{5(x+3) + 2(x+5)}{(x-7)(x+5)(x+3)}$$ 5. **Expand the numerator:** $$5(x+3) + 2(x+5) = 5x + 15 + 2x + 10 = 7x + 25$$ 6. **Write the expression:** $$\frac{7x + 25}{(x-7)(x+5)(x+3)}$$ 7. **Check if numerator can be factored:** The numerator $7x + 25$ cannot be factored further since 7 and 25 have no common factors. 8. **Final simplified expression in factored form:** $$\frac{7x + 25}{(x-7)(x+5)(x+3)}$$ This is the simplified form with the denominator fully factored and numerator simplified.