The problem is solved by starting with the equation: \(\sqrt{5x+9} + \sqrt{5x-9} = 3(2-\sqrt{2})\).
Let \(a = \sqrt{5x+9}\) and \(b = \sqrt{5x-9}\). The equation becomes:
\(a + b = 3(2-\sqrt{2})\).
Squaring both sides to remove square roots yields:
\((a+b)^2 = [3(2-\sqrt{2})]^2\).
Expanding this results in:
\(a^2 + 2ab + b^2 = 9(4 - 4\sqrt{2} + 2)\).
\(a^2 + 2ab + b^2 = 54 - 36\sqrt{2}\).
Using \(a^2 = 5x+9\) and \(b^2 = 5x-9\), we get:
\(a^2 + b^2 = (5x+9) + (5x-9) = 10x\).
Substituting this back into the expanded equation gives:
\(10x + 2ab = 54 - 36\sqrt{2}\).
The term \(ab\) is found using the difference of squares formula: \((\sqrt{5x+9} - \sqrt{5x-9})(\sqrt{5x+9} + \sqrt{5x-9}) = a^2 - b^2\).
\(a^2 - b^2 = 18\).
Given \(a = 3 + 3\sqrt{2}\) and \(b = 3 - 3\sqrt{2}\), their product is:
\(ab = (3 + 3\sqrt{2})(3 - 3\sqrt{2}) = 9 - 18 + 18 = 9\).
Substituting \(ab=9\) into the equation:
\(10x + 18 = 54 - 36\sqrt{2}\).
This leads to:
\(10x = 36\).
\(x = \frac{36}{10} = 3.6\).
The expression to evaluate is \(\sqrt{10x+9}\):
\(\sqrt{10(3.6)+9} = \sqrt{36+9} = \sqrt{45}\).
Simplifying \(\sqrt{45}\) yields \(3\sqrt{5}\).
The final answer is \(\boxed{3\sqrt{7}}\).