\(\dfrac{1}{x-5x^2}+\dfrac{25x-15}{25x^2-1}\)
\(=\dfrac{1}{x\left(1-5x\right)}+\dfrac{25x-15}{\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x+x\left(25x-15\right)}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x+25x^2-15}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1-10x+25x^2}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{\left(1-5x\right)^2}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1-5x}{x\left(1+5x\right)}\)
\(\dfrac{1}{x-5x^2}-\dfrac{25x-15}{25x^2-1}\)
\(=\dfrac{1}{x-5x^2}+\dfrac{25x-15}{1-25x^2}\)
\(=\dfrac{1}{x\left(1-5x\right)}+\dfrac{25x-15}{\left(1-5x\right)\left(1+5x\right)}\) MTC: \(x\left(1-5x\right)\left(1+5x\right)\)
\(=\dfrac{1+5x}{x\left(1-5x\right)\left(1+5x\right)}+\dfrac{x\left(25x-15\right)}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x+x\left(25x-15\right)}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1+5x+25x^2-15x}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{25x^2-10x+1}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{\left(5x-1\right)^2}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{\left(1-5x\right)^2}{x\left(1-5x\right)\left(1+5x\right)}\)
\(=\dfrac{1-5x}{x\left(1+5x\right)}\)
