\(=\dfrac{x^2+4x+3+x^2-4x+3-2x\left(x-1\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{2x^2+6-2x^2+2x}{\left(x-3\right)\left(x+3\right)}=\dfrac{2}{x-3}\)
\(\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}-\dfrac{2x\left(1-x\right)}{9-x^2}\)
\(=\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}-\dfrac{2x\left(1-x\right)}{\left(3-x\right)\left(3+x\right)}\)
\(=\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}+\dfrac{2x\left(1-x\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\dfrac{\left(1-x\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}+\dfrac{2x\left(1-x\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{x^2+3x+x+3}{\left(x-3\right)\left(x+3\right)}-\dfrac{x-3-x^2+3x}{\left(x-3\right)\left(x+3\right)}+\dfrac{2x-2x^2}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{x^2+3x+x+3-x+3+x^2-3x+2x-2x^2}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{2x+6}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{2\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}\)
\(=\dfrac{2}{x-3}\)
\(\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}-\dfrac{2x\left(1-x\right)}{9-x^2}=\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}+\dfrac{2x\left(1-x\right)}{x^2-9}=\dfrac{x+1}{x-3}-\dfrac{1-x}{x+3}+\dfrac{2x\left(1-x\right)}{\left(x-3\right)\left(x+3\right)}=\dfrac{\left(x+1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\dfrac{\left(1-x\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}+\dfrac{2x\left(1-x\right)}{\left(x-3\right)\left(x+3\right)}=\dfrac{x^2+4x+3}{\left(x-3\right)\left(x+3\right)}-\dfrac{-x^2+4x-3}{\left(x-3\right)\left(x+3\right)}+\dfrac{2x-2x^2}{\left(x-3\right)\left(x+3\right)}=\dfrac{\left(x^2+4x+3\right)-\left(-x^2+4x-3\right)+\left(2x-2x^2\right)}{\left(x-3\right)\left(x+3\right)}=\dfrac{x^2+4x+3+x^2-4x+3+2x-2x^2}{\left(x-3\right)\left(x+3\right)}=\dfrac{2x+6}{\left(x-3\right)\left(x+3\right)}=\dfrac{2\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}=\dfrac{2}{x-3}\)
