^{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 x^{2}-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...x^{2}-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 x^{3}-4x^{2}-3x/LCD.(remember to multiply by negative 1)
...x^{2}+2/x^{2}-x-2 goes in the LCD (x-2)(x+1) times so multiply the numerator x^{2}+2 times the remaining factor x+3.
we get. x^{3}+3x^{2}+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)
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