Answer
$\displaystyle \frac{2(2x^{2}+5x-2)}{(x-2)(x+2)(x+3)}$
Work Step by Step
Step 1:
Factor each denominator
$ x^{2}-4=\qquad$ ... a difference of squares,
$=(x-2)(x+2)$
For $x^{2}+bx+c$, we search for factors of c whose sum is b:
$ x^{2}+x-6=\qquad$ ... +3 and -2
$=(x+3)(x-2)$
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 = $(x-2)(x+2)(x+3)$
Step 3:
Write each rational expression using the LCM as the denominator. Simplify.
$\displaystyle \frac{4x}{(x-2)(x+2)}\cdot\frac{(x+3)}{(x+3)}-\frac{2}{(x+3)(x-2)}\frac{(x+2)}{(x+2)}$
$=\displaystyle \frac{4x^{2}+12x-(2x+4)}{(x-2)(x+2)(x+3)}$
$=\displaystyle \frac{4x^{2}+10x-4}{(x-2)(x+2)(x+3)}$
$=\displaystyle \frac{2(2x^{2}+5x-2)}{(x-2)(x+2)(x+3)}$
For $ax^{2}+bx+c$ in the numerator,
we search for factors of ac whose sum is b, and rewrite the $bx$ term:
... no two factors of -4 add to +5, so we leave it as it is.
