a) \(\dfrac{2x}{x-4}+\dfrac{8}{4-x}=\dfrac{2x}{x-4}-\dfrac{8}{x-4}=\dfrac{2x-8}{x-4}\)
\(=\dfrac{2\left(x-4\right)}{x-4}=2\)
c, \(\dfrac{x^2}{x-y}+\dfrac{y^2}{y-x}=\dfrac{x^2}{x-y}-\dfrac{y^2}{x-y}=\dfrac{x^2-y^2}{x-y}\)
\(=\dfrac{\left(x-y\right)\left(x+y\right)}{x-y}=x+y\)
b) \(\dfrac{x+1}{4x}+\dfrac{2x-1}{5x}+\dfrac{4x+3}{20x}\)
\(=\dfrac{5\left(x+1\right)}{20x}+\dfrac{4\left(2x-1\right)}{20x}+\dfrac{4x+3}{20x}\)
\(=\dfrac{5x+5+8x-4+4x+3}{20x}\)
\(=\dfrac{17x+4}{20x}\)
c) \(\dfrac{x}{2x+4}+\dfrac{2x+2}{x^2+2x}=\dfrac{x}{2\left(x+2\right)}+\dfrac{2x+2}{x\left(x+2\right)}\)
\(=\dfrac{x^2}{2x\left(x+2\right)}+\dfrac{2\left(2x+2\right)}{2x\left(x+2\right)}=\dfrac{x^2+4x+4}{2x\left(x+2\right)}\)
\(=\dfrac{\left(x+2\right)^2}{2x\left(x+2\right)}=\dfrac{x+2}{2x}\)
d) \(\dfrac{x^2+1}{x^2+2x+1}+\dfrac{1-x}{x+1}+\dfrac{2x}{x^2+2x+1}\)
\(=\dfrac{x^2+1+2x}{x^2+2x+1}+\dfrac{1-x}{x+1}=1+\dfrac{1-x}{x+1}\)
\(=\dfrac{x+1+1-x}{x+1}=\dfrac{2}{x+1}\)
\(\text{#}Toru\)
