\(\dfrac{\dfrac{2x+1}{x-1}+\dfrac{2x-1}{x+1}}{\dfrac{4x^2+1}{x^2-1}}\)
\(=\dfrac{2x+1}{x-1}+\dfrac{2x-1}{x+1}:\dfrac{4x^2+1}{x^2-1}\)
\(=\dfrac{2x+1}{x-1}+\dfrac{2x-1}{x+1}.\dfrac{x^2-1}{4x^2+1}\)
\(=\dfrac{\left(2x+1\right)\left(x+1\right)+\left(2x-1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}.\dfrac{\left(x-1\right)\left(x+1\right)}{4x^2+1}\)
\(=\dfrac{2x^2+2x+x+1+2x^2-2x-x+1}{\left(x-1\right)\left(x+1\right)}.\dfrac{\left(x-1\right)\left(x+1\right)}{4x^2+1}\)
\(=\dfrac{\left(4x^2+2\right)\left(x-1\right)\left(x+1\right)}{\left(4x^2+1\right)\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{4x^2+2}{4x^2+1}\).
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