1/x+3 - x/x-2 + x^2+2/x^2-x-2 (used super/subscript and "/" to show fraction. Not sure if that's proper format)
first factor x2-x-2 to get (x-2)(x+1). The reason I factored it was to determine what's going to be my LCD. With that being said my LCD therefore is: LCD: (x+3)(x-2)(x+1). Notice that (x-2) was a repeated factor but you only account for it once.
...1/x+3 goes into the LCD (x+3) times so you multiply the numerator which is 1 by the remaining factors (x-2)(x+1). we get (x+2)(x+1)/LCD...x2-x-2/LCD
...x/x-2 goes in the LCD (x-2) ties so u multiply the numerator x by (x+3)(x+1). we get x(x+3)(x+1). multiply everything out to get x3-4x2-3x/LCD.(remember to multiply by negative 1)
...x2+2/x2-x-2 goes in the LCD (x-2)(x+1) times so multiply the numerator x2+2 times the remaining factor x+3.
we get. x3+3x2+2x+6.
Now that we have found the LCD we can perform the indicated operations. so the final answer after collecting like terms is...-2x+4/LCD.. the cubic and quadratic factors will cancel out.
now in your numerator, divide it by -2 to get x-2.
so we no have x-2/(x+3)(x+1)(x-2)
Notice that (x-2) cancels so the final answer is ......................1/(x+3)(x+1)