# How do you simplify (x^2+8x+16)/(x+2)div(x^2+6x+8)/(x^2-4)?

Mar 4, 2018

The factored expression is $\frac{{x}^{2} + 2 x - 8}{x + 2}$.

#### Explanation:

Here's the strategy:

First, factor all the polynomials. Then, cancel any common terms. Next, change the division of two fractions to multiplication, and flip the second fraction. Lastly, multiply the fractions and cancel any factors in common.

$\textcolor{w h i t e}{=} \frac{{x}^{2} + 8 x + 16}{x + 2} \div \frac{{x}^{2} + 6 x + 8}{{x}^{2} - 4}$

$= \frac{\left(x + 4\right) \left(x + 4\right)}{x + 2} \div \frac{\left(x + 2\right) \left(x + 4\right)}{\left(x - 2\right) \left(x + 2\right)}$

$= \frac{{\left(x + 4\right)}^{2}}{x + 2} \div \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(x + 2\right)}}} \left(x + 4\right)}{\left(x - 2\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(x + 2\right)}}}}$

$= \frac{{\left(x + 4\right)}^{2}}{x + 2} \div \frac{x + 4}{x - 2}$

$= \frac{{\left(x + 4\right)}^{2}}{x + 2} \times \frac{x - 2}{x + 4}$

$= \frac{{\left(x + 4\right)}^{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}} \left(x - 2\right)}{\left(x + 2\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(x + 4\right)}}}}$

$= \frac{\left(x + 4\right) \left(x - 2\right)}{x + 2}$

$= \frac{{x}^{2} + 2 x - 8}{x + 2}$

Here's what the graph looks like:

graph{(x^2+2x-8)/(x+2) [-30.54, 30.53, -28.27, 28.27]}

Mar 4, 2018

$\frac{\left(x + 4\right) \left(x - 2\right)}{\left(x + 2\right)}$

#### Explanation:

$\frac{{x}^{2} + 8 x + 16}{x + 2} \div \frac{{x}^{2} + 6 x + 8}{{x}^{2} - 4}$

To divide by a fraction, multiply by the reciprocal:

$\frac{\textcolor{b l u e}{\left({x}^{2} + 8 x + 16\right)}}{x + 2} \times \frac{\textcolor{red}{{x}^{2} - 4}}{\textcolor{g r e e n}{\left({x}^{2} + 6 x + 8\right)}}$

You cannot cancel with fractions unless there are factors.

Factorise wherever possible:

$\frac{\textcolor{b l u e}{\left(x + 4\right) \left(x + 4\right)}}{x + 2} \times \frac{\textcolor{red}{\left(x + 2\right) \left(x - 2\right)}}{\textcolor{g r e e n}{\left(x + 4\right) \left(x + 2\right)}}$

Cancel common factors

$\frac{\textcolor{b l u e}{\cancel{\left(x + 4\right)} \left(x + 4\right)}}{\cancel{\left(x + 2\right)}} \times \frac{\textcolor{red}{\cancel{\left(x + 2\right)} \left(x - 2\right)}}{\textcolor{g r e e n}{\cancel{\left(x + 4\right)} \left(x + 2\right)}}$

$= \frac{\left(x + 4\right) \left(x - 2\right)}{\left(x + 2\right)}$