What is the sum? \( \frac{3 y}{y^{2}+7 y+10}+\frac{2}{y+2} \) \( \frac{5}{y-5} \) \( \frac{5(y+2)}{(y-2)(y+5)} \) \( \frac{5}{y+5} \) \( \frac{5(y-2)}{(y-5)(y+2)} \)

To find the sum of the fractions given, we first need to simplify each term and combine them over a common denominator. 1. The first fraction can be rewritten: \[\frac{3y}{(y+2)(y+5)} + \frac{2}{y+2}\] We can get a common denominator: \[\frac{3y + 2(y+5)}{(y+2)(y+5)} = \frac{3y + 2y + 10}{(y+2)(y+5)} = \frac{5y + 10}{(y+2)(y+5)} = \frac{5(y+2)}{(y+2)(y+5)}\] Now, it simplifies to \(\frac{5}{y+5}\). 2. Combine with the second term \(\frac{5}{y-5}\): The common denominator is \((y+5)(y-5)\), hence: \[\frac{5(y-5)}{(y+5)(y-5)} + \frac{5(y+5)}{(y-5)(y+5)}\] This results in: \[\frac{5(y-5 + y+5)}{(y+5)(y-5)} = \frac{5(2y)}{(y+5)(y-5)} = \frac{10y}{(y+5)(y-5)}\] 3. Now, we proceed with the fraction \(\frac{5(y-2)}{(y-5)(y+2)}\): To combine it with \(\frac{10y}{(y+5)(y-5)}\), we need to find the common denominator which would be \((y+5)(y-5)(y+2)\). 4. However, we also include the last term: \[\frac{5(y-2)}{(y-5)(y+2)}\] The sum of all terms can be organized and simplified through algebraic combination into a single fraction. **Final Sum:** The complete calculation will yield a single combined expression after factoring and common denominators are accounted for. Each term undergoes simplification akin to assembling a jigsaw, leading to an answer that encapsulates the essence of all contributions, while also respecting variable conditions. Thus, the overall quest to find the sum leads us to articulate sums in manageable mathematical expressions gracefully!