a/ $\dfrac{4x+1}{2}-\dfrac{3x+2}{3}\\=\dfrac{3(4x+1)}{6}-\dfrac{2(3x+2)}{6}\\=\dfrac{12x+3-6x-4}{6}\\=\dfrac{6x-1}{6}$

b/ $\dfrac{x+3}{x}-\dfrac{x}{x-3}+\dfrac{9}{x²-3x}\\=\dfrac{(x+3)(x-3)}{x(x-3)}-\dfrac{x²}{x(x-3)}+\dfrac{9}{x(x-3)}=\dfrac{x²-9-x²+9}{x(x-3)}\\=\dfrac{0}{x(x-3)}=0$

c/ $\dfrac{x+3}{x²-1}-\dfrac{1}{x²+x}\\=\dfrac{x+3}{(x+1)(x-1)}-\dfrac{1}{x(x+1)}\\=\dfrac{x(x+3)}{x(x+1)(x-1)}-\dfrac{x-1}{x(x+1)(x-1)}\\=\dfrac{x²+3x-x+1}{x(x+1)(x-1)}\\=\dfrac{x²+2x+1}{x(x+1)(x-1)}\\=\dfrac{(x+1)²}{x(x+1)(x-1)}\\=\dfrac{x+1}{x(x-1)}$

d/ $\dfrac{1}{3x-2}-\dfrac{4}{3x+2}-\dfrac{-10x+8}{9x²-4}\\=\dfrac{3x+2}{(3x+2)(3x-2)}-\dfrac{4(3x-2)}{(3x+2)(3x-2)}-\dfrac{-10x+8}{(3x-2)(3x+2)}\\=\dfrac{3x+2-12x+8+10x-8}{(3x+2)(3x-2)}\\=\dfrac{x+2}{(3x+2)(3x-2)}$

e/ $\dfrac{3}{2x²+2x}+\dfrac{2x-1}{x²-1}-\dfrac{2}{x}\\=\dfrac{3}{2x(x+1)}+\dfrac{2x-1}{(x-1)(x+1)}-\dfrac{2}{x}\\=\dfrac{3(x-1)}{2x(x+1)(x-1)}+\dfrac{2x(2x-1)}{2x(x+1)(x-1)}-\dfrac{4(x²-1)}{2x(x+1)(x-1)}\\=\dfrac{3x-3+4x²-2x-4x²+4}{2x(x+1)(x-1)}\\=\dfrac{x+1}{2x(x+1)(x-1)}\\=\dfrac{1}{2x(x-1)}$

f/ $\dfrac{3x}{5x+5y}-\dfrac{x}{10x-10y}\\=\dfrac{3x}{5(x+y)}-\dfrac{x}{10(x-y)}\\=\dfrac{6x(x-y)}{10(x+y)(x-y)}-\dfrac{x(x+y)}{10(x+y)(x-y)}\\=\dfrac{6x²-6xy-x²-xy}{10(x+y)(x-y)}\\=\dfrac{5x²-7xy}{10(x+y)(x-y)}$

g/ $\dfrac{4a²-3a+5}{a³-1}-\dfrac{1-2a}{a²+a+1}-\dfrac{6}{a-1}\\=\dfrac{4a²-3a+5}{(a-1)(a²+a+1)}-\dfrac{(1-2a)(a-1)}{(a-1)(a²+a+1)}-\dfrac{6(a²+a+1)}{(a-1)(a²+a+1)}\\=\dfrac{4a²-3a+5}{(a-1)(a²+a+1)}-\dfrac{2a²+3a-1}{(a-1)(a²+a+1)}-\dfrac{6a²+6a+6}{(a-1)(a²+a+1)}\\=\dfrac{4a²-3a+5+2a²-3a+1-6a²-6a-6}{(a-1)(a²+a+1)}\\=\dfrac{-12a}{(a-1)(a²+a+1)}$

h/ $\dfrac{5x²-y²}{xy}-\dfrac{3x-2y}{y}\\=\dfrac{5x²-y²}{xy}-\dfrac{x(3x-2y)}{xy}\\=\dfrac{5x²-y²-3x²+2xy}{xy}\\=\dfrac{2x²+2xy-y²}{xy}$

i/ $\dfrac{x+9y}{x²-9y²}-\dfrac{3y}{x²+3xy}\\=\dfrac{x+9y}{(x-3y)(x+3y)}-\dfrac{3y}{x(x+3y)}\\=\dfrac{x(x+9y)}{x(x-3y)(x+3y)}-\dfrac{3y(x-3y)}{x(x-3y)(x+3y)}\\=\dfrac{x²+9xy-3xy+9y²}{x(x+3y)(x-3y)}\\=\dfrac{x²-6xy+9y²}{x(x+3y)(x-3y)}\\=\dfrac{(x-3y)²}{x(x-3y)(x+3y)}=\dfrac{x-3y}{x(x+3y)}$

k/ $\dfrac{3x+2}{x²-2x+1}-\dfrac{6}{x²-1}-\dfrac{3x-2}{x²+2x+1}\\=\dfrac{3x+2}{(x-1)²}-\dfrac{6}{(x+1)(x-1)}-\dfrac{3x-2}{(x+1)²}\\=\dfrac{(3x+2)(x+1)²}{[(x+1)(x-1)]²}-\dfrac{6(x+1)(x-1)}{[(x+1)(x-1)]²}-\dfrac{(3x-2)(x-1)²}{[(x+1)(x-1)]²}\\=\dfrac{3x³+8x²+7x+2-6x²+6-3x³+8x²-7x+2}{[(x+1)(x-1)]²}\\=\dfrac{10x²+10}{[(x+1)(x-1)]²}$

l/ $\dfrac{3}{2x+6}-\dfrac{x-6}{2x²+6x}\\=\dfrac{3}{2x+6}-\dfrac{x-6}{x(2x+6)}\\=\dfrac{3x}{x(2x+6)}-\dfrac{x-6}{x(2x+6)}\\=\dfrac{3x-x+6}{x(2x+6)}\\=\dfrac{2x+6}{x(2x+6)}\\=\dfrac{1}{x}$

m/ $x²+1-\dfrac{x^4+1}{x²+1}\\=\dfrac{(x²+1)²}{x²+1}-\dfrac{x^4+1}{x²+1}\\=\dfrac{x^4+2x²+1-x^4-1}{x²+1}\\=\dfrac{2x²}{x²+1}$

n/ $\dfrac{5}{a+1}-\dfrac{10}{a-(a²+1)}-\dfrac{15}{a³+1}\\=\dfrac{5}{a+1}-\dfrac{-10}{a²-a+1}-\dfrac{15}{(a+1)(a²-a+1)}\\=\dfrac{5(a²-a+1)}{(a+1)(a²-a+1)}-\dfrac{-10(a+1)}{(a+1)(a²-a+1)}-\dfrac{15}{(a+1)(a²-a+1)}\\=\dfrac{5a²-5a+5+10a+10-15}{(a+1)(a²-a+1)}\\=\dfrac{5a²+5a}{(a+1)(a²-a+1)}\\=\dfrac{5a(a+1)}{(a+1)(a²-a+1)}\\=\dfrac{5a}{a²-a+1}$

`a)(4x+1)/2-(3x+2)/3`

`=(3(4x+1))/6-(2(3x+2))/6`

`=(12x+3-6x-4)/6`

`=(6x-1)/6`

`b)(x+3)/x-x/(x-3)+9/(x^2-3x)`

`=((x+3)(x-3))/(x(x-3))-x^2/(x(x-3))+9/(x(x-3))`

`=((x+3)(x-3)-x^2+9)/(x(x-3))`

`=(x^2-9-x^2+9)/(x(x-3))`

`=0`

`c)(x+3)/(x^2-1)-1/(x^2+x)`

`=(x+3)/((x-1)(x+1))-1/(x(x+1))`

`=(x(x+3)-(x-1))/(x(x-1)(x+1))`

`=(x^2-2x+1)/(x(x-1)(x+1)`

`=(x+1)/(x(x-1))`

`d)1/(3x-2)-4/(3x+2)-(-10x+8)/(9x^2-4)`

`=1/(3x−2)−4/(3x+2)​−(-2(5x−4)​)/(9x^2−4)`

`=1/(3x−2)−4/(3x+2​)+(2(5x−4)​)/((3x+2)(3x−2))`

`=(3x+2−12x+8+10x−8​)/((3x+2)(3x−2))`

`=(3x+2−12x+8+10x−8)/((3x−2)(3x+2))`

`=(x+2)/((3x+2)(3x−2))`

`e)3/(2x^2+2x)+(2x-1)/(x^2-1)-2/x`

`=3/(2x(x+1)​)+(2x−1)/(​(x+1)(x−1))−2/​x`

`=(3x−3+4x2−2x−4x^2+4x−4x+4)/(2x(x+1)(x−1))`

`=(x+1​)/(2x(x+1)(x−1))`

`=1/(2x(x−1))`

`f)(3x)/(5x+5y)-x/(10x-10y)`

`=(3​x)/(5(x+y))−x/(​10(x−y))`

`=(30x(x−y)−5x(x+y))/(50(x+y)(x−y))`

`=(5x(5x−7y))/(​50(x+y)(x−y))`

`=(x(5x−7y))/(​10(x+y)(x−y))`

`g)(4a^2-3a+5)/(a^3-1)-(1-2a)/(a^2+a+1)-6/(a-1)`

`=(4a^2−3a+5)/((a−1)(a^2+a+1)​)−(1−2a)/(a^2+a+1)​−6/(a−1​)`

`=(4a^2−3a+5−a+1+2a^2−2a−6a^2−6a−6​)/((a−1)(a2+a+1))`

`=(−12a​)/((a−1)(a^2+a+1))`

`h)(5x^2-y^2)/(xy)-(3x-2y)/y`

`=(5x^2−y^2−3x^2+2xy​)/(xy)`

`=(2x^2−y^2+2xy)/(xy)`

`i)(x+9y)/(x^2-9y^2)-(3y)/(x^2+3xy)`

`=(x+9y)/(​(x+3y)(x−3y))−(3y)/(​x(x+3y))`

`=(x^2+9xy−3yx+9y^2​)/(x(x+3y)(x−3y))`

`=((x+3y)^2​)/(x(x+3y)(x−3y))`

`k)(3x+2)/(x^2-2x+1)-6/(x^2-1)-(3x-2)/(x^2+2x+1)`

`=(3x+2)/((x−1)^2)−6/((x+1)(x−1)​)−(3x−2)/(​(x+1)^2)`

`=((3x+2)(x+1)^2 -6(x-1)(x+1)-(3x-2)(x-1)^2 ​ )/((x-1)^2 (x+1)^2)`

`=(3x^3+8x^2+7x+2-6x^2+6-3x^3+8x^2-7x+2)/((x-1)^2 (x+1)^2)`

`=(10x^2+10)/((x-1)^2 (x+1)^2)`

`l)3/(2x+6)-(x-6)/(2x^2+6x)`

`=3/(​2(x+3))−(x−6)/(​2x(x+3))`

`=(3x−x+6)/(​2x(x+3))`

`=2/(2x)`

`=1/x`

`m)x^2+1-(x^4+1)/(x^2+1)`

`=((x^2+1)^2)/(x^2+1)-(x^4+1)/(x^2+1)`

`=(x^4+2x^2+1-x^4-1)/(x^2+1)`

`=(2x^2)/(x^2+1)`

`n)5/(a+1)-10/(a-(a^2+1))-15/(a^3+1)`

`=(5(a^2-a+1))/(a^3+1)+(10(a+1))/(a^3+1)-15/(a^3+1)`

`=(5a^2-5a+5+10a+10-15)/(a^3+1)`

`=(5a^2+5a)/((a+1)(a^2-a+1))`

`=(5a(a+1))/((a+1)(a^2-a+1))`

`=(5a)/(a^2-a+1)`