Start with the equation \(2(3^{2x-1}) = 4^{x+1}\).

Take the logarithm of both sides:

\(\ln(2) + (2x-1)\ln(3) = (x+1)\ln(4)\).

Rearrange to form a linear equation:

\((2x-1)\ln(3) - x\ln(4) = \ln(4) - \ln(2)\).

Simplify and solve for \(x\):

\(x = \frac{\ln(4) - \ln(2) + \ln(3)}{2\ln(3) - \ln(4)}\).

Calculate the value of \(x\) to 2 decimal places:

\(x \approx 2.21\).