b) \(\dfrac{2}{x^2+2x}+\dfrac{8-2x}{x^3+8}=\dfrac{2}{x\left(x+2\right)}+\dfrac{2}{\left(x+2\right)\left(x^2-2x+4\right)}\)\(=\dfrac{2\left(x^2-2x+4\right)}{x\left(x+2\right)\left(x^2-2x+4\right)}+\dfrac{2x}{x\left(x+2\right)\left(x^2-2x+4\right)}\)

\(=\dfrac{2x^2-4x+8+2x}{x\left(x+2\right)\left(x^2-2x+4\right)}=\dfrac{2x^2-2x+8}{x\left(x+2\right)\left(x^2-2x+4\right)}\)

c) \(\dfrac{x}{1-x^3}+\dfrac{1}{x^2+x-2}=\dfrac{-x}{x^3-1}+\dfrac{1}{x^2+2x-x-2}\)

\(=\dfrac{-x}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{x\left(x+2\right)-\left(x+2\right)}\)

\(=\dfrac{-x}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{\left(x-1\right)\left(x-2\right)}\)

\(=\dfrac{-x\left(x-2\right)}{\left(x-1\right)\left(x-2\right)\left(x^2+x+1\right)}+\dfrac{1\left(x^2+x+1\right)}{\left(x-1\right)\left(x-2\right)\left(x^2+x+1\right)}\)\(=\dfrac{-x^2+2x+x^2+x+1}{\left(x-1\right)\left(x-2\right)\left(x^2+x+1\right)}\)

\(=\dfrac{3x+1}{\left(x-1\right)\left(x-2\right)\left(x^2+x+1\right)}\)

f) \(\dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{3x-2}{2x-4x^2}\)

\(=\dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{2-3x}{2x\left(2x-1\right)}\)

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

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

\(=\dfrac{-2x+1}{2x\left(2x-1\right)}\)

\(=\dfrac{-\left(2x-1\right)}{2x\left(2x-1\right)}=\dfrac{-1}{2x}\)

h) \(\dfrac{1}{2x^2-x-1}+\dfrac{1}{6x^2+9x+3}\)

\(=\dfrac{1}{2x^2+x-2x-1}+\dfrac{1}{3\left(2x^2+3x+1\right)}\)

\(=\dfrac{1}{x\left(2x+1\right)-\left(2x+1\right)}+\dfrac{1}{3\left(2x^2+x+2x+1\right)}\)

\(=\dfrac{1}{\left(2x+1\right)\left(x-1\right)}+\dfrac{1}{3\left[x\left(2x+1\right)+\left(2x+1\right)\right]}\)

\(=\dfrac{1}{\left(2x+1\right)\left(x-1\right)}+\dfrac{1}{3\left(x+1\right)\left(2x+1\right)}\)

\(=\dfrac{3x+3+x-1}{3\left(2x+1\right)\left(x+1\right)\left(x-1\right)}\)

\(=\dfrac{4x+2}{3\left(2x+1\right)\left(x+1\right)\left(x-1\right)}=\dfrac{2}{3\left(x+1\right)\left(x-1\right)}\)

e) \(\dfrac{x}{x+2}+\dfrac{4}{x-2}+\dfrac{8x}{4-x^2}\)

\(=\dfrac{x}{x+2}+\dfrac{4}{x-2}+\dfrac{-8x}{x^2-2^2}\)

\(=\dfrac{x}{x+2}+\dfrac{4}{x-2}+\dfrac{-8x}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{x^2-2x+4x+8-8x}{\left(x-2\right)\left(x+2\right)}=\dfrac{x^2-6x+8}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{x^2-2x-4x+8}{\left(x-2\right)\left(x+2\right)}=\dfrac{x\left(x-2\right)-4\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{\left(x-2\right)\left(x-4\right)}{\left(x-2\right)\left(x+2\right)}=\dfrac{x-4}{x+2}\)