Answer
$\displaystyle \frac{x^{3}-2x^{2}+4x+3}{x^{2}(x+1)(x-1)}$
Work Step by Step
Step 1:
Factor each denominator
$x^{2}+x=x(x+1)$
$x^{3}-x^{2}=x^{2}(x-1)$
Step 2: The LCM is the product of each of these factors raised to a power equal to the greatest number of times that the factor occurs in the polynomials.
LCM = $x^{2}(x+1)(x-1)$
Step 3:
Write each rational expression using the LCM as the denominator. Simplify.
$\displaystyle \frac{1\cdot x(x+1)(x-1)}{x\cdot x(x+1)(x-1)}-\frac{2\cdot x(x-1)}{x\cdot x(x+1)(x-1)}+\frac{3(x+1)}{x^{2}(x-1)(x+1)}$
$=\displaystyle \frac{x(x^{2}-1)-(2x^{2}-2x)+3x+3}{x^{2}(x+1)(x-1)}$
$=\displaystyle \frac{x^{3}-x-2x^{2}+2x+3x+3}{x^{2}(x+1)(x-1)}$
$=\displaystyle \frac{x^{3}-2x^{2}+4x+3}{x^{2}(x+1)(x-1)}$
