=\(\dfrac{x}{x.\left(x-5\right)}\)+ \(\dfrac{x+1}{4.\left(x+5\right)}\)
= \(\dfrac{4x}{4x\left(x-5\right)}\) + \(\dfrac{x\left(x+1\right)}{4x\left(x-5\right)}\)
= \(\dfrac{4x+x^2+x}{4x\left(x-5\right)}\) = \(\dfrac{5x^3}{4x\left(x-5\right)}\)
\(\dfrac{x}{x^2-5x}+\dfrac{x+1}{4x+20}=\dfrac{x}{x\left(x-5\right)}+\dfrac{x+1}{4\left(x+5\right)}\)
\(=\dfrac{x\cdot4\left(x+5\right)}{4x\left(x-5\right)\left(x+5\right)}+\dfrac{\left(x+1\right)\cdot x\left(x-5\right)}{4\cdot x\left(x-5\right)\left(x+5\right)}\)
\(=\dfrac{4x\left(x+5\right)}{4x\left(x-5\right)\left(x+5\right)}+\dfrac{\left(x+1\right)\left(x^2-5x\right)}{4x\left(x-5\right)\left(x+5\right)}\)
\(=\dfrac{4x\left(x+5\right)+\left(x+1\right)\left(x^2-5x\right)}{4x\left(x+5\right)\left(x-5\right)}\)
\(=\dfrac{4x^2+20x+x^3-5x^2+x^2-5x}{4x\left(x+5\right)\left(x-5\right)}\)
\(=\dfrac{x^3+15x}{4x\left(x+5\right)\left(x-5\right)}=\dfrac{x\left(x^2+15\right)}{4x\left(x+5\right)\left(x-5\right)}=\dfrac{x^2+15}{4\left(x+5\right)\left(x-5\right)}=\dfrac{x^2+15}{4\left(x^2-25\right)}=\dfrac{x^2+15}{4x^2-100}\)
