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

Rewrite \(3^{x+2}\) as \(3^x imes 3^2\), giving \(3^x imes 9 = 3^x + 9\).

Factor out \(3^x\) from the left side: \(3^x (9 - 1) = 9\).

This simplifies to \(3^x imes 8 = 9\).

Divide both sides by 8: \(3^x = \frac{9}{8}\).

Take the logarithm of both sides: \(x \ln 3 = \ln \left(\frac{9}{8}\right)\).

Solve for \(x\): \(x = \frac{\ln \left(\frac{9}{8}\right)}{\ln 3}\).

Calculate \(x\) to 3 significant figures: \(x \approx 0.107\).