\(\dfrac{12x^2+28x+8}{9x^2-1}=\dfrac{4\left(3x^2+7x+2\right)}{9x^2-1}=\dfrac{4\left(3x+1\right)\left(x+2\right)}{\left(3x-1\right)\left(3x+1\right)}=\dfrac{4x+8}{3x-1}\)
\(\dfrac{12x^2+28x+8}{9x^2-1}=\dfrac{12x^2+4x+24x+8}{\left(3x-1\right)\left(3x+1\right)}\)
\(=\dfrac{4x\left(3x+1\right)+8\left(3x+1\right)}{\left(3x-1\right)\left(3x+1\right)}=\dfrac{\left(3x+1\right)\left(4x+8\right)}{\left(3x-1\right)\left(3x+1\right)}\)
\(=\dfrac{4x+8}{3x-1}\)
\(\dfrac{12x^2+28x+8}{9x^2-1}\left(x\ne\pm\dfrac{1}{3}\right)\)
\(=\dfrac{4\left(3x^2+7x+2\right)}{\left(3x\right)^2-1^2}\)
\(=\dfrac{4\left(3x^2+x+6x+2\right)}{\left(3x-1\right)\left(3x+1\right)}\)
\(=\dfrac{4\left[x\left(3x+1\right)+2\left(3x+1\right)\right]}{\left(3x-1\right)\left(3x+1\right)}\)
\(=\dfrac{4\left(3x+1\right)\left(x+2\right)}{\left(3x+1\right)\left(3x-1\right)}\)
\(=\dfrac{4\left(x+2\right)}{3x-1}\)
\(=\dfrac{4x+8}{3x-1}\)
