To find the sum of the given fractions, we first need to factorize the denominators where possible and then combine the fractions into a single rational expression before simplifying. Let's walk through this step-by-step.
The given expression is:
[tex]\[
\frac{3 y}{y^2+7 y+10}+\frac{2}{y+2}
\][/tex]
First, let's factorize the quadratic expression in the denominator \(y^2 + 7y + 10\). We look for two numbers that multiply to 10 (the constant term) and add up to 7 (the coefficient of the linear term \(y\)):
[tex]\[
y^2 + 7y + 10 = (y + 2)(y + 5)
\][/tex]
So, the expression now is:
[tex]\[
\frac{3 y}{(y + 2)(y + 5)} + \frac{2}{y + 2}
\][/tex]
Next, we need a common denominator to combine these fractions. The least common denominator (LCD) will be \((y + 2)(y + 5)\).
Rewrite each fraction with the common denominator:
[tex]\[
\frac{3 y}{(y + 2)(y + 5)} + \frac{2(y + 5)}{(y + 2)(y + 5)}
\][/tex]
Now, combine the numerators over the common denominator:
[tex]\[
\frac{3y + 2(y + 5)}{(y + 2)(y + 5)}
\][/tex]
Distribute and simplify the numerator:
[tex]\[
3y + 2(y + 5) = 3y + 2y + 10 = 5y + 10
\][/tex]
So our fraction becomes:
[tex]\[
\frac{5y + 10}{(y + 2)(y + 5)}
\][/tex]
Observe that the numerator \(5y + 10\) can be factored out:
[tex]\[
5y + 10 = 5(y + 2)
\][/tex]
Thus, the fraction simplifies to:
[tex]\[
\frac{5(y + 2)}{(y + 2)(y + 5)}
\][/tex]
The \((y + 2)\) terms in the numerator and denominator cancel out:
[tex]\[
\frac{5}{y + 5}
\][/tex]
Therefore, the simplified sum of the given fractions is:
[tex]\[
\frac{5}{y + 5}
\][/tex]
The correct answer is:
[tex]\[
\boxed{\frac{5}{y+5}}
\][/tex]
