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To solve the equation \( x^2 + 9 = 45 \), we first isolate \( x^2 \):
\[
x^2 + 9 = 45
\]
Subtract 9 from both sides:
\[
x^2 = 45 - 9
\]
\[
x^2 = 36
\]
Next, we take the square root of both sides:
\[
x = \pm \sqrt{36}
\]
This gives us:
\[
x = 6 \quad \text{or} \quad x = -6
\]
Thus, the solution set is \( \{-6, 6\} \).
Amelia is the only one who correctly identifies the solution set as \( \{-6, 6\} \).
Therefore, the correct answer is:
**Amelia**