# How do you simplify and find the restrictions for ( (x^2 -x-12)/( x^2 -2x-15) ) / ( (x^2 + 8x+12)/( x^2 -5x-14))?

Oct 5, 2016

The expression simplifies to $\frac{\left(x - 4\right) \left(x + 7\right)}{\left(x - 5\right) \left(x + 6\right)}$ with restrictions being $x \ne 5 , - 3 , - 6 , 7 \mathmr{and} - 2$.

#### Explanation:

=(((x - 4)(x + 3))/((x - 5)(x + 3)))/(((x + 6)(x + 2))/((x - 7)(x + 2))

$= \frac{\frac{x - 4}{x - 5}}{\frac{x + 6}{x + 7}}$

$= \frac{\left(x - 4\right) \left(x + 7\right)}{\left(x - 5\right) \left(x + 6\right)}$

Restrictions occur when the denominator of the original expression equal $0$. We can determine restrictions by setting the denominator to $0$ and solving.

$\left(x - 5\right) \left(x + 3\right) = 0$

$x = 5 \mathmr{and} - 3$

AND

$\left(x + 6\right) \left(x + 2\right) = 0$

$x = - 6 \mathmr{and} - 2$

AND

$\left(x - 7\right) \left(x + 2\right) = 0$

$x = 7 \mathmr{and} - 2$

Hopefully this helps!