$a) \dfrac{4}{x+2}+\dfrac{2}{x-2}+\dfrac{5x-6}{4-x^2} ( x \neq ± 2)$
$=\dfrac{4(x-2)}{(x+2)(x-2)}+\dfrac{2(x+2)}{(x+2)(x-2)}-\dfrac{5x-6}{(x+2)(x-2)}$
$=\dfrac{4x-8+2x+4-5x+6}{(x+2)(x-2)}$
$=\dfrac{x+2}{(x+2)(x-2)}=\dfrac{1}{x-2}$
$b) \dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{3x-2}{2x-4x^2}$ $\left ( x \neq 0 ; x \neq \dfrac{1}{2} \right )$
$=\dfrac{(1-3x)(2x-1)}{2x(2x-1)}+\dfrac{(3x-2)2x}{2x(2x-1)}-\dfrac{3x-2}{2x(2x-1)}$
$=\dfrac{2x-1-6x^2+3x+6x^2-4x-3x+2}{2x(2x-1)}$
$=\dfrac{-2x+1}{2x(2x-1)}=\dfrac{-1}{2x}$
$c) \dfrac{1}{x^2+6x+9}+\dfrac{1}{6x-x^2-9}+\dfrac{x}{x^2-9}$ $(x \neq ±3)$
$=\dfrac{(x-3)^2}{(x+3)^2(x-3)^2}-\dfrac{(x+3)^2}{(x+3)^2(x-3)^2}+\dfrac{x(x^2-9)}{(x+3)^2(x-3)^2}$
$=\dfrac{x^2-6x+9-x^2-6x-9+x^3-9x}{(x+3)^2(x-3)^2}$
$=\dfrac{x^3-21x}{(x+3)^2(x-3)^2}$
$d) \dfrac{x^2+2}{x^3-1}+\dfrac{2}{x^2+x+1}+\dfrac{1}{1-x} ( x \neq 1 )$
$=\dfrac{x^2+2+2(x-1)-1(x^2+x+1)}{(x-1)(x^2+x+1)}$
$=\dfrac{x^2+2+2x-2-x^2-x-1}{(x-1)(x^2+x+1)}$
$=\dfrac{x-1}{(x-1)(x^2+x+1)}$
$=\dfrac{1}{x^2+x+1}$
$e) \dfrac{x}{x+2y}+\dfrac{x}{x-2y}+\dfrac{4xy}{4y^2-x^2}$
$=\dfrac{x(x-2y)+x(x+2y)-4xy}{(x-2y)(x+2y)}$
$=\dfrac{x^2-2xy+x^2+2xy-4xy}{(x-2y)(x+2y)}$
$=\dfrac{2x(x-2y)}{(x-2y)(x+2y)}$
$=\dfrac{2x}{x+2y}$
$f) \dfrac{4x^2-3x+17}{x^3-1}+\dfrac{2x-1}{x^2+x+1}+\dfrac{6}{1-x} ( x \neq 1)$
$=\dfrac{4x^2-3x+17+(2x-1)(x-1)-6(x^2+x+1)}{(x-1)(x^2+x+1)}$
$=\dfrac{4x^2-3x+17+2x^2-2x-x+1-6x^2-6x-6}{(x-1)(x^2+x+1)}$
$=\dfrac{-12x+12}{(x-1)(x^2+x+1)}$
$=\dfrac{-12(x-1)}{(x-1)(x^2+x+1)}$
$=\dfrac{-12}{x^2+x+1}$
