\(\frac{4x^4-20x^3+13x^2+30x+9}{\left(4x^2-1\right)^2}\)
\(=\frac{4x^3\left(x-3\right)-8x^2\left(x-3\right)-11x\left(x-3\right)-3\left(x-3\right)}{\left(4x^2-1\right)^2}\)
\(=\frac{\left(x-3\right)\left(4x^3-8x^2-11x-3\right)}{\left(4x^2-1\right)^2}\)
\(=\frac{\left(x-3\right)\left[4x^2\left(x-3\right)+4x\left(x-3\right)+\left(x-3\right)\right]}{\left[\left(2x-1\right)\left(2x+1\right)\right]^2}\)
\(=\frac{\left(x-3\right)^2\left(4x^2+4x+1\right)}{\left(2x-1\right)^2\left(2x+1\right)^2}=\frac{\left(x-3\right)^2\left(2x+1\right)^2}{\left(2x-1\right)^2\left(2x+1\right)^2}=\frac{\left(x-3\right)^2}{\left(2x-1\right)^2}\)
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