Giải thích các bước giải:

1.ĐKXĐ: $x\ne\pm3y$

Ta có:

$K=\dfrac{x+9y}{x^2-9y^2}-\dfrac{3y}{x^2+3xy}$

$\to K=\dfrac{x+9y}{(x-3y)(x+3y)}-\dfrac{3y}{x(x+3y)}$

$\to K=\dfrac{\left(x+9y\right)x}{x\left(x-3y\right)\left(x+3y\right)}-\dfrac{3y\left(x-3y\right)}{x\left(x-3y\right)\left(x+3y\right)}$

$\to K=\dfrac{x^2+6xy+9y^2}{x\left(x-3y\right)\left(x+3y\right)}$

$\to K=\dfrac{(x+3y)^2}{x(x-3y)(x+3y)}$

$\to K=\dfrac{x+3y}{x\left(x-3y\right)}$

2.ĐKXĐ: $x\ne1, x\ne 0$

Ta có:

$I=\dfrac{4x^2-3x+5}{x^3-1}-\dfrac{1-2x}{x^2+x+1}+\dfrac{6x}{x-x^2}$

$\to I=\dfrac{4x^2-3x+5}{(x-1)(x^2+x+1)}+\dfrac{2x-1}{x^2+x+1}-\dfrac{6x}{x(x-1)}$

$\to I=\dfrac{4x^2-3x+5+\left(2x-1\right)\left(x-1\right)-6\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}$

$\to I=\dfrac{-12x}{\left(x-1\right)\left(x^2+x+1\right)}$

3.ĐKXĐ: $x\ne -1$

Ta có:

$K=\dfrac5{x+1}-\dfrac{10}{x-(x^2+1)}-\dfrac{15}{x^3+1}$

$\to K=\dfrac5{x+1}+\dfrac{10}{x^2-x+1}-\dfrac{15}{(x+1)(x^2-x+1)}$

$\to K=\dfrac{5\left(x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}+\dfrac{10\left(x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}-\dfrac{15}{\left(x+1\right)\left(x^2-x+1\right)}$

$\to K=\dfrac{5\left(x^2-x+1\right)+10\left(x+1\right)-15}{\left(x+1\right)\left(x^2-x+1\right)}$

$\to K=\dfrac{5x^2+5x}{\left(x+1\right)\left(x^2-x+1\right)}$

$\to K=\dfrac{5x(x+1)}{\left(x+1\right)\left(x^2-x+1\right)}$

$\to K=\dfrac{5x}{x^2-x+1}$

4.ĐKXĐ: $x\ne\pm3$

Ta có:

$M=\dfrac{6x}{x^2-9}-\dfrac{5x}{3-x}+\dfrac{x}{x+3}$

$\to M=\dfrac{6x}{(x-3)(x+3)}+\dfrac{5x}{x-3}+\dfrac{x}{x+3}$

$\to M=\dfrac{6x}{\left(x-3\right)\left(x+3\right)}+\dfrac{5x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}+\dfrac{x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}$

$\to M=\dfrac{6x+5x\left(x+3\right)+x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}$

$\to M=\dfrac{6x^2+18x}{\left(x-3\right)\left(x+3\right)}$

$\to M=\dfrac{6x(x+3)}{\left(x-3\right)\left(x+3\right)}$

$\to M=\dfrac{6x}{x-3}$

5.Ta có:

$P=\dfrac{\sqrt{x}}{\sqrt{x}-2}:(\dfrac{x-2}{x-4}-\dfrac1{\sqrt{x}+2})$

$\to P=\dfrac{\sqrt{x}}{\sqrt{x}-2}:\dfrac{x-2-(\sqrt{x}-2)}{(\sqrt{x}-2)(\sqrt{x}+2)}$

$\to P=\dfrac{\sqrt{x}}{\sqrt{x}-2}:\dfrac{x-\sqrt{x}}{(\sqrt{x}-2)(\sqrt{x}+2)}$

$\to P=\dfrac{\sqrt{x}}{\sqrt{x}-2}\cdot \dfrac{(\sqrt{x}-2)(\sqrt{x}+2)}{x-\sqrt{x}}$

$\to P=\dfrac{\sqrt{x}}{\sqrt{x}-2}\cdot \dfrac{(\sqrt{x}-2)(\sqrt{x}+2)}{\sqrt{x}(\sqrt{x}-1)}$

$\to P=\dfrac{\sqrt{x}+2}{\sqrt{x}-1}$