Question

1. \\(\frac{3}{y + 5} + \frac{y}{y^2 + 7y + 10}\\)

Explanation:

Step1: Factor the denominator

First, factor the quadratic denominator \(y^2 + 7y + 10\). 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 expression becomes \(\frac{3}{y + 5}+\frac{y}{(y + 2)(y + 5)}\).

Step2: Find a common denominator

The common denominator of \(\frac{3}{y + 5}\) and \(\frac{y}{(y + 2)(y + 5)}\) is \((y + 2)(y + 5)\). Rewrite \(\frac{3}{y + 5}\) with the common denominator:
\(\frac{3}{y + 5}=\frac{3(y + 2)}{(y + 2)(y + 5)}\) (multiply numerator and denominator by \(y + 2\)).

Step3: Add the fractions

Now add the two fractions:
\[

$$\begin{align*} \frac{3(y + 2)}{(y + 2)(y + 5)}+\frac{y}{(y + 2)(y + 5)}&=\frac{3(y + 2)+y}{(y + 2)(y + 5)}\\ &=\frac{3y + 6 + y}{(y + 2)(y + 5)}\\ &=\frac{4y + 6}{(y + 2)(y + 5)} \end{align*}$$

\]

Step4: Simplify the numerator (optional)

We can factor the numerator \(4y + 6 = 2(2y + 3)\), so the expression is \(\frac{2(2y + 3)}{(y + 2)(y + 5)}\) or we can leave it as \(\frac{4y + 6}{y^2 + 7y + 10}\) (by expanding the denominator \((y + 2)(y + 5)=y^2 + 7y + 10\)).

Answer:

\(\frac{4y + 6}{(y + 2)(y + 5)}\) (or \(\frac{2(2y + 3)}{(y + 2)(y + 5)}\) or \(\frac{4y + 6}{y^2 + 7y + 10}\))