# How do you simplify (4y + 8 )/(y^2 + 7y + 10)?

##### 1 Answer
Sep 8, 2015

$\frac{4}{y + 5}$

#### Explanation:

Your starting expression is

$\frac{4 y + 8}{{y}^{2} + 7 y + 10}$

Notice that you can factor the numerator like this

$4 y + 8 = 4 \cdot \left(y + 2\right)$

Now focus on finding a way to factor the denominator. To do that, find the rotts to this quadratic equation by using the quadratic formula

${y}^{2} + 7 y + 10 = 0$

${y}_{1 , 2} = \frac{- 7 \pm \sqrt{{7}^{2} - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

${y}_{1 , 2} = \frac{- 7 \pm \sqrt{3}}{2} = \left\{\begin{matrix}{y}_{1} = \frac{- 7 - 3}{2} = - 5 \\ {y}_{2} = \frac{- 7 + 3}{2} = - 2\end{matrix}\right.$

This means that the denominator can be factored as

${y}^{2} + 7 y + 10 = \left[y - \left(- 2\right)\right] \cdot \left[y - \left(- 5\right)\right]$

$= \left(y + 2\right) \left(y + 5\right)$

Your starting expression can now be written as

$\frac{4 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(y + 2\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(y + 2\right)}}} \cdot \left(y + 5\right)} = \textcolor{g r e e n}{\frac{4}{y + 5}}$