Start with the equation: \(2\log_2 x = 3 + \log_2(x + 1)\).

Apply the power rule of logarithms: \(\log_2 x^2 = 3 + \log_2(x + 1)\).

Use the property \(\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)\):

\(\log_2 \left(\frac{x^2}{x+1}\right) = 3\).

Convert the logarithmic equation to an exponential equation:

\(\frac{x^2}{x+1} = 2^3 = 8\).

Multiply both sides by \(x+1\):

\(x^2 = 8(x + 1)\).

Expand and rearrange to form a quadratic equation:

\(x^2 - 8x - 8 = 0\).

Solve the quadratic equation using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), \(c = -8\).

Calculate the discriminant: \(b^2 - 4ac = (-8)^2 - 4(1)(-8) = 64 + 32 = 96\).

\(x = \frac{8 \pm \sqrt{96}}{2}\).

\(x = \frac{8 \pm 4\sqrt{6}}{2}\).

\(x = 4 \pm 2\sqrt{6}\).

Calculate the positive root: \(x = 4 + 2\sqrt{6} \approx 8.90\).

Thus, the solution is \(x = 8.90\).