1. **State the problem:** Solve the equation $\left(2e\right)^{x^2} - 3 = e^9$ for $x$. 2. **Rewrite the equation:** Add 3 to both sides to isolate the exponential term: $$\left(2e\right)^{x^2} = e^9 + 3$$ 3. **Express the base in terms of $e$:** Since $2e = 2 \times e$, rewrite the left side as: $$\left(2e\right)^{x^2} = \left(2 \cdot e\right)^{x^2} = 2^{x^2} \cdot e^{x^2}$$ 4. **Rewrite the equation:** $$2^{x^2} \cdot e^{x^2} = e^9 + 3$$ 5. **Divide both sides by $e^{x^2}$:** $$2^{x^2} = \frac{e^9 + 3}{e^{x^2}} = e^{9 - x^2} + 3e^{-x^2}$$ This is complicated to solve algebraically, so instead, consider the original equation and take natural logarithm after isolating the exponential term. 6. **Return to step 2 and isolate:** $$\left(2e\right)^{x^2} = e^9 + 3$$ 7. **Take natural logarithm on both sides:** $$\ln\left(\left(2e\right)^{x^2}\right) = \ln\left(e^9 + 3\right)$$ 8. **Use logarithm power rule:** $$x^2 \ln(2e) = \ln\left(e^9 + 3\right)$$ 9. **Simplify $\ln(2e)$:** $$\ln(2e) = \ln 2 + \ln e = \ln 2 + 1$$ 10. **Solve for $x^2$:** $$x^2 = \frac{\ln\left(e^9 + 3\right)}{\ln 2 + 1}$$ 11. **Take square root:** $$x = \pm \sqrt{\frac{\ln\left(e^9 + 3\right)}{\ln 2 + 1}}$$ **Final answer:** $$x = \pm \sqrt{\frac{\ln\left(e^9 + 3\right)}{\ln 2 + 1}}$$