How do I solve $3^{\ln{2}} \times x^{\ln x + \ln 6 + 1} = \frac{3e^2}{4}$

Question

I've been going at this question for 2 hours, my teacher wants us to solve for x without a graphing calculator.

\begin{equation} 3^{\ln{2}} \times x^{\ln x + \ln 6 + 1} = \frac{3e^2}{4} \end{equation}

Answer 1

Hint: isolate the $x$ term.

Hint 2: identify a quadratic in $\ln x$.

$$(\ln x+\ln 6+1)\ln x = \ln 3+\ln e^2-\ln 4-\ln 3^{\ln 2}$$