Answer
$-\dfrac{2x(x^2-2)}{(x+2)(x^2-x-3)}$
Work Step by Step
Simplify the numerator $\frac{x-2}{x+2}+\frac{x-1}{x+1}$.
The LCD is $(x+2)(x+1)$.
Make the expresions simillar by multiplying $x+1$ to both the numerator and the denominator of the first expression and $x+2$ to the second expression to obtain:
$=\dfrac{(x-2)(x+1)}{(x+2)(x+1)}+\dfrac{(x-1)(x+2)}{(x+1)(x+2)}$
$=\dfrac{(x-2)(x+1)+(x-1)(x+2)}{(x+2)(x+1)}$
Use FOIL method.
$=\dfrac{x^2+x-2x-2+x^2+2x-x-2}{(x+2)(x+1)}$
Simplify.
$=\dfrac{2x^2-4}{(x+2)(x+1)}$
Factor.
$=\dfrac{2(x^2-2)}{(x+2)(x+1)}$
Simplify the denominator $\frac{x}{x+1}-\frac{2x-3}{x}$
The LCD is $x(x+1)$.
Make the expressions similar by multiplying $x$ to both the numerator and the denominator of the first expression and $x+1$ to the second expression to obtain:
$=\dfrac{x^2}{x(x+1)}-\dfrac{(2x-3)(x+1)}{x(x+1)}$
$=\dfrac{x^2-(2x-3)(x+1)}{x(x+1)}$
Use FOIL method.
$=\dfrac{x^2-(2x^2+2x-3x-3)}{x(x+1)}$
$=\dfrac{x^2-2x^2-2x+3x+3}{x(x+1)}$
$=\dfrac{-x^2+x+3}{x(x+1)}$
Factor.
$=\dfrac{-(x^2-x-3)}{x(x+1)}$
Substitute back into the given expression.
$=\dfrac{\dfrac{2(x^2-2)}{(x+2)(x+1)}}{\dfrac{-(x^2-x-3)}{x(x+1)}}$
Use the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ to obtain:
.
$=\dfrac{2(x^2-2)}{(x+2)(x+1)}\cdot \dfrac{x(x+1)}{-(x^2-x-3)}$
Cancel common factors:
$=-\dfrac{2x(x^2-2)}{(x+2)(x^2-x-3)}$
Hence, the lowest term is:
$-\dfrac{2x(x^2-2)}{(x+2)(x^2-x-3)}$
