a: \(\dfrac{2x+5}{5x^2y^2}+\dfrac{8}{5xy^2}+\dfrac{2x-1}{x^2y^2}\)
\(=\dfrac{2x+5+8x+5\left(2x-1\right)}{5x^2y^2}\)
\(=\dfrac{10x+5+10x-5}{5x^2y^2}=\dfrac{20x}{5x^2y^2}=\dfrac{4}{xy^2}\)
b: \(\dfrac{4x^2-3x+5}{x^3-1}-\dfrac{1-2x}{x^2+x+1}-\dfrac{6}{x-1}\)
\(=\dfrac{4x^2-3x+5}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{2x-1}{x^2+x+1}-\dfrac{6}{x-1}\)
\(=\dfrac{4x^2-3x+5+\left(2x-1\right)\left(x-1\right)-6\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{4x^2-3x+5+2x^2-3x+1-6x^2-6x-6}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{-12x}{x^3-1}\)
c: \(\dfrac{x^4+4x^2+5}{5x^3+5}\cdot\dfrac{2x}{x^2+4}\cdot\dfrac{3x^3+3}{x^4+4x^2+5}\)
\(=\dfrac{2x}{5x^3+5}\cdot\dfrac{3x^3+3}{x^2+4}\)
\(=\dfrac{2x}{5\left(x^3+1\right)}\cdot\dfrac{3\left(x^3+1\right)}{x^2+2}=\dfrac{6x}{5\left(x^2+2\right)}\)
d: \(\dfrac{5x+1}{2x-3}\cdot\dfrac{x+2}{25x^2-1}-\dfrac{8-3x}{25x^2-1}\cdot\dfrac{5x+1}{2x-3}\)
\(=\dfrac{\left(5x+1\right)\left(x+2\right)+\left(3x-8\right)\left(5x+1\right)}{\left(2x-3\right)\left(25x^2-1\right)}\)
\(=\dfrac{\left(5x+1\right)\left(x+2+3x-8\right)}{\left(2x-3\right)\left(5x-1\right)\left(5x+1\right)}\)
\(=\dfrac{4x-6}{\left(2x-3\right)\left(5x-1\right)}=\dfrac{2}{5x-1}\)
