Question

How do you simplify `(x^2+4x-45)/(x^2+10x+9)` and find the restrictions on the variable?

Answer 1

`(x-5)/(x+1)` provided `x!=-9` and `x!=-1`

Explanation:

`(color(blue)(x^2+4x-45))/(color(red)(x^2+10x+9))`

`color(white)("XXX")=(color(blue)((x+9)(x-5)))/(color(red)((x+9)(x+1)))`

Provide `(x+9)!=0` (that is provided `x!=-9`)
[since division by zero is undefined]
we can divide the numerator and denominator by `(x+9)` to get:
`color(white)("XXX")=(color(blue)(x-5))/(color(red)(x+1))`

which is defined provided
`color(white)("XXX")x+1!=0` (or expressed differently, provided `x!=-1)`