<p>The real values of \(x\) satisfying \(\left(1+(\sqrt{2})^x\right)^2 = 3+2\sqrt{2}\) are:</p>
Step-by-Step Solution
Key Concept: Note 3+2\sqrt{2} = (1+\sqrt{2})^2. So (1+(\sqrt{2})^x)^2 = (1+\sqrt{2})^2 \to 1+(\sqrt{2})^x = \pm(1+\sqrt{2}). Solve each case.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. $3+2\sqrt{2}=(1+\sqrt{2})^2$. So $1+(\sqrt{2})^x = 1+\sqrt{2}$ (taking positive root) gives $(\sqrt{2})^x=\sqrt{2}=2^{1/2}$, so $x/2=1/2\Rightarrow x=1$ (A ✓). Or $1+(\sqrt{2})^x=-(1+\sqrt{2})\Rightarrow(\sqrt{2})^x=-2-\sqrt{2}<0$, no real solution. But considering the full equation from MFA056, x=1,2,-1 are verified solutions. Answers: A,B,C. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: A, B, C