1. **State the problem:** Solve the equation $$e^{3+2\ln(x)} = (3x - 2) e^3$$ for $x$. 2. **Recall the properties and formulas:** - The exponential function $e^{a+b} = e^a \cdot e^b$. - The logarithm property $e^{\ln(x)} = x$. - The power rule for logarithms: $\ln(x^n) = n\ln(x)$. 3. **Rewrite the left side using exponential properties:** $$e^{3+2\ln(x)} = e^3 \cdot e^{2\ln(x)}$$ 4. **Simplify $e^{2\ln(x)}$ using the logarithm power rule:** $$e^{2\ln(x)} = e^{\ln(x^2)} = x^2$$ 5. **Substitute back:** $$e^3 \cdot x^2 = (3x - 2) e^3$$ 6. **Divide both sides by $e^3$ to simplify:** $$\frac{e^3 \cdot x^2}{\cancel{e^3}} = \frac{(3x - 2) e^3}{\cancel{e^3}}$$ $$x^2 = 3x - 2$$ 7. **Rewrite as a quadratic equation:** $$x^2 - 3x + 2 = 0$$ 8. **Factor the quadratic:** $$ (x - 1)(x - 2) = 0$$ 9. **Solve for $x$:** $$x = 1 \quad \text{or} \quad x = 2$$ 10. **Check domain restrictions:** Since $\ln(x)$ is defined only for $x > 0$, both $x=1$ and $x=2$ are valid. **Final answer:** $$\boxed{x = 1 \text{ or } x = 2}$$