a: \(=\dfrac{\left(x^2+1\right)\left(x^2-1\right)-x^4-1}{x^2-1}\)
\(=\dfrac{x^4-1-x^4-1}{x^2-1}=-\dfrac{2}{x^2-1}\)
b: \(=\dfrac{\left(x+y\right)^2-x^2-y^2}{x+y}\)
\(=\dfrac{x^2+2xy+y^2-x^2-y^2}{x+y}=\dfrac{2xy}{x+y}\)
c: \(=\dfrac{\left(x+1\right)\left(x+3\right)+\left(2x-1\right)\left(x-3\right)+x\left(1-x\right)}{x^2-9}\)
\(=\dfrac{x^2+4x+3+2x^2-7x+3+x-x^2}{x^2-9}\)
\(=\dfrac{2x^2-2x+6}{x^2-9}\)
d: \(\dfrac{5}{2x^2+6x}-\dfrac{4-3x^2}{x^2-9}-3\)
\(=\dfrac{5}{2x\left(x+3\right)}+\dfrac{3x^2-4}{\left(x-3\right)\left(x+3\right)}-3\)
\(=\dfrac{5\left(x-3\right)+2x\left(3x^2-4\right)-3\cdot2x\left(x^2-9\right)}{2x\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{5x-15+6x^3-8x-6x^3+54x}{2x\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{51x-15}{2x\left(x+3\right)\left(x-3\right)}\)
e: \(=\dfrac{3x+2}{\left(x-1\right)^2}-\dfrac{6}{\left(x-1\right)\left(x+1\right)}-\dfrac{3x-2}{\left(x+1\right)^2}\)
\(=\dfrac{\left(3x+2\right)\left(x+1\right)^2-6\left(x^2-1\right)-\left(3x-2\right)\left(x-1\right)^2}{\left(x-1\right)^2\cdot\left(x+1\right)^2}\)
\(=\dfrac{\left(3x+2\right)\left(x+1\right)\left(x+1\right)-\left(3x-2\right)\left(x-1\right)\left(x-1\right)-6\left(x^2-1\right)}{\left(x-1\right)^2\cdot\left(x+1\right)^2}\)
\(=\dfrac{\left(3x^2+5x+2\right)\left(x+1\right)-\left(3x^2-5x+2\right)\left(x-1\right)-6\left(x^2-1\right)}{\left(x-1\right)^2\cdot\left(x+1\right)^2}\)
\(=\dfrac{3x^3+8x^2+7x+2-3x^3+8x^2-7x+2-6x^2+6}{\left(x-1\right)^2\left(x+1\right)^2}\)
\(=\dfrac{10x^2+10}{\left(x-1\right)^2\left(x+1\right)^2}\)
