Question

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

Explanation:

Step1: Factor the denominator

Factor \( y^2 + 7y + 10 \). We need two numbers that multiply to 10 and add to 7, which are 2 and 5. So \( y^2 + 7y + 10=(y + 2)(y + 5) \). The first fraction becomes \( \frac{3y}{(y + 2)(y + 5)} \), and the second fraction is \( \frac{2}{y + 2} \).

Step2: Find a common denominator

The common denominator of \( (y + 2)(y + 5) \) and \( y + 2 \) is \( (y + 2)(y + 5) \). Rewrite the second fraction with the common denominator: \( \frac{2}{y + 2}=\frac{2(y + 5)}{(y + 2)(y + 5)} \).

Step3: Add the fractions

Now add the two fractions: \( \frac{3y}{(y + 2)(y + 5)}+\frac{2(y + 5)}{(y + 2)(y + 5)}=\frac{3y+2(y + 5)}{(y + 2)(y + 5)} \).

Step4: Simplify the numerator

Simplify the numerator: \( 3y+2(y + 5)=3y + 2y+10 = 5y + 10=5(y + 2) \). So the fraction becomes \( \frac{5(y + 2)}{(y + 2)(y + 5)} \).

Step5: Cancel common factors

Cancel the common factor \( (y + 2) \) (assuming \( y
eq - 2 \)): \( \frac{5(y + 2)}{(y + 2)(y + 5)}=\frac{5}{y + 5} \).

Answer:

\(\frac{5}{y + 5}\) (corresponding to the third option, e.g., C. \(\frac{5}{y + 5}\) if options are labeled C for this one)