Answer
$\dfrac{5}{x^{2}-4}+\dfrac{2}{x^{2}-4x+4}-\dfrac{3}{x^{2}-x-6}=\dfrac{4x^{2}-15x+6}{(x-2)^{2}(x+2)(x-3)}$
Work Step by Step
$\dfrac{5}{x^{2}-4}+\dfrac{2}{x^{2}-4x+4}-\dfrac{3}{x^{2}-x-6}$
Factor the three rational expressions completely:
$\dfrac{5}{(x-2)(x+2)}+\dfrac{2}{(x-2)^{2}}-\dfrac{3}{(x-3)(x+2)}=...$
Evaluate the indicated operations and simplify:
$...=\dfrac{5(x-3)(x-2)+2(x+2)(x-3)-3(x-2)^{2}}{(x-2)^{2}(x+2)(x-3)}=...$
$...=\dfrac{5x^{2}-25x+30+2x^{2}-2x-12-3x^{2}+12x-12}{(x-2)^{2}(x+2)(x-3)}=...$
$...=\dfrac{4x^{2}-15x+6}{(x-2)^{2}(x+2)(x-3)}$
