Question

How do you combine `(x+1)/(x^2+6x+8) + (x-4)/(x^2-3x-10)`?

Answer 1

Convert the two terms so they have a common denominator then add the numerators (and possibly simplify)

Since `x^2+6x+8 = color(red)((x+2))(x+4)`
and `x^2-3x-10 = color(red)((x+2))(x-5)`
the obvious candidate for denominator is
`(x+2)(x+4)(x-5)`

and
`(x+1)/(x^2+6x+8) + (x-4)/(x^2-3x-10)`

can be written as

`((x+1)(x-5) +(x-4)(x+4))/((x+2)(x+4)(x-5))`

`=( x^2 -4x -5 +x^2-16)/((x+2)(x+4)(x-5))`

`= (2x^2-4x-21)/((x+2)(x+4)(x-5))`

(with no obvious factors of the numerator)
`= (2x^2-4x-21)/(x^3+x^2-22x-40`