# Question f78ff

Mar 13, 2017

$\frac{{y}^{2} + 11 y + 30}{y + 5} = y + 6$

#### Explanation:

Given: $\frac{{y}^{2} + 11 y + 30}{y + 5}$

Can be solved by reducing the numerator (top bracket) to see how it will react with the denominator (bottom bracket).

We have $\left({y}^{2} + 11 y + 30\right)$ that we know can be factored into two terms containing $y$ because the $y$ exponent is $2$.

(y....... ) (y....... )

Looking at the 30, we see it can be factored into $2 \times 15$; or $10 \times 3$; or $5 \times 6$.
The last factors look appealing because they will add up to $11$ to match the given middle term.

Now: (y....... $5$) (y$\ldots \ldots .$6)

And we know both signs here must be positive because we had to $a \mathrm{dd}$ $5$ to $6$ to get the $11$.

Then: (y + 5) (y + 6)#

Placing our factors into the original equation:

$\frac{\left(y + 5\right) \left(y + 6\right)}{y + 5}$

Gives us good news, because one of the factors cancels out the denominator to leave:

$1 \left(y + 6\right) = \left(y + 6\right)$

$\frac{{y}^{2} + 11 y + 30}{y + 5} = y + 6$