Question

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

Explanation:

Step1: Factor the denominator

Factor $y^{2}+7y + 10$ as $(y + 2)(y+5)$. So the expression becomes $\frac{3y}{(y + 2)(y + 5)}+\frac{2}{y + 2}$.

Step2: Find a common - denominator

The common denominator of the two fractions is $(y + 2)(y + 5)$. Rewrite $\frac{2}{y + 2}$ as $\frac{2(y + 5)}{(y + 2)(y + 5)}$.

Step3: Add the 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

Expand $3y+2(y + 5)$ to get $3y+2y+10=5y + 10 = 5(y + 2)$. So the fraction is $\frac{5(y + 2)}{(y + 2)(y + 5)}$.

Step5: Cancel out the common factor

Cancel out the common factor $(y + 2)$ (assuming $y
eq - 2$). The result is $\frac{5}{y + 5}$.

Answer:

$\frac{5}{y + 5}$