a: \(\dfrac{4}{x}=\dfrac{4\left(x+2\right)}{x\left(x+2\right)}=\dfrac{4x+8}{x\left(x+2\right)};\dfrac{5}{x^2+2x}=\dfrac{5}{x\left(x+2\right)}\)
b: \(\dfrac{x-2}{x^2-6x+9}=\dfrac{x-2}{\left(x-3\right)^2}=\dfrac{\left(x-2\right)\cdot\left(x+3\right)}{\left(x+3\right)\left(x-3\right)^2}=\dfrac{x^2+x-6}{\left(x+3\right)\left(x-3\right)^2}\)
\(\dfrac{x+1}{x^2-9}=\dfrac{x+1}{\left(x-3\right)\left(x+3\right)}=\dfrac{\left(x+1\right)\left(x-3\right)}{\left(x-3\right)^2\cdot\left(x+3\right)}=\dfrac{x^2-2x-3}{\left(x-3\right)^2\cdot\left(x+3\right)}\)
c: \(\dfrac{5}{x^2-1}=\dfrac{5}{\left(x-1\right)\cdot\left(x+1\right)}=\dfrac{5x}{x\left(x-1\right)\left(x+1\right)}\)
\(\dfrac{2}{x^2-x}=\dfrac{2}{x\left(x-1\right)}=\dfrac{2\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}=\dfrac{2x+2}{x\left(x-1\right)\left(x+1\right)}\)
d: \(\dfrac{2x}{x+5}=\dfrac{2x\left(x-5\right)}{\left(x+5\right)\left(x-5\right)}=\dfrac{2x^2-10x}{\left(x+5\right)\left(x-5\right)}\)
\(\dfrac{10}{x^2-25}=\dfrac{10}{\left(x-5\right)\left(x+5\right)}\)
e: \(\dfrac{x^3}{x^2-4}=\dfrac{x^3}{\left(x-2\right)\left(x+2\right)}=\dfrac{x^3\cdot\left(x-2\right)\cdot\left(x-3\right)}{\left(x-2\right)^2\cdot\left(x+2\right)\left(x-3\right)}\)
\(\dfrac{x^2}{x^2-4x+4}=\dfrac{x^2}{\left(x-2\right)^2}=\dfrac{x^2\cdot\left(x+2\right)\left(x-3\right)}{\left(x-2\right)^2\cdot\left(x+2\right)\left(x-3\right)}\)
\(\dfrac{x}{x^2-5x+6}=\dfrac{x}{\left(x-2\right)\left(x-3\right)}=\dfrac{x\cdot\left(x-2\right)\left(x+2\right)}{\left(x-2\right)^2\cdot\left(x+2\right)\left(x-3\right)}\)
