Question 151424
{{{(x-3)/(x^2-2x-15)+(x+2)/(5-x)}}} Start with the given expression



{{{(x-3)/((x+3)(x-5))+(x+2)/(5-x)}}} Factor {{{x^2-2x-15}}} to get {{{(x+3)(x-5)}}}



{{{(x-3)/((x+3)(x-5))+(x+2)/(-1(x-5))}}} Factor {{{5-x}}} to get {{{-1(x-5)}}}



{{{(x-3)/((x+3)(x-5))+(-x-2)/(x-5)}}} Simplify.



Now let's find the LCM of the denominators {{{(x+3)(x-5)}}} and {{{(x-5)}}}. It turns out that the LCM of these denominators is {{{(x+3)(x-5)}}}. So the goal is to get both denominators to the LCD {{{(x+3)(x-5)}}}



{{{(x-3)/((x+3)(x-5))+((x+3)/(x+3))((-x-2)/((x-5)))}}} Multiply the 2nd fraction by {{{((x+3))/((x+3))}}}



{{{(x-3)/((x+3)(x-5))+((x+3)(-x-2))/((x+3)(x-5)))}}} Combine the fractions.



{{{(x-3)/((x+3)(x-5))+(-x^2-2x-3x-6)/((x+3)(x-5)))}}} FOIL



{{{(x-3)/((x+3)(x-5))+(-x^2-5x-6)/((x+3)(x-5)))}}} Combine like terms.



{{{(x-3-x^2-5x-6)/((x+3)(x-5)))}}} Add the fractions.



{{{(-x^2-4x-9)/((x+3)(x-5)))}}} Combine like terms.



{{{(-x^2-4x-9)/(x^2-2x-15)}}} FOIL the denominator.



So {{{(x-3)/(x^2-2x-15)+(x+2)/(5-x)}}} adds and simplifies to {{{(-x^2-4x-9)/(x^2-2x-15)}}} 



In other words, {{{(x-3)/(x^2-2x-15)+(x+2)/(5-x)=(-x^2-4x-9)/(x^2-2x-15)}}} where {{{x<>-3}}} or {{{x<>5}}}