Start your solution by cross-multiplying,

\(\frac{\sqrt{2}}{x+\sqrt{2}}=\frac{1}{x-\sqrt{2}}\)

[x - √2] √2  = x + √2

where √2×√2=2 

x√2 - √2 * √2 = x + √2

then collect like terms

x√2 - x = √2 + 2

and factorize accordingly to get the unknown.

x(√2 - 1) = √2 + 2

x = \(\frac{√2 + 2}{√2 - 1}\)

rationalize 

x = \(\frac{√2 + 2}{√2 - 1}\) * \(\frac{√2 + 1}{√2 + 1}\)

x = \(\frac{√4 + √2 + 2√2 + 2}{√4  + √2 - √2 - 1}\)

x = \(\frac{2 + 3√2 + 2}{2 - 1}\)

x = \(\frac{3√2 + 4}{1}\)

x = 3√2 + 4