To solve the equation \((1\sqrt{2}x - 3)(9 - \sqrt{2}x + 1) = 0\), we need to set each factor equal to zero and solve for \(x\).

First, let's simplify the equation:

\[
(1\sqrt{2}x - 3)(9 - \sqrt{2}x + 1) = 0
\]

This can be written as:

\[
(\sqrt{2}x - 3)(10 - \sqrt{2}x) = 0
\]

Now, we set each factor equal to zero:

1. \(\sqrt{2}x - 3 = 0\)
2. \(10 - \sqrt{2}x = 0\)

### Solving the first equation:

\[
\sqrt{2}x - 3 = 0
\]

Add 3 to both sides:

\[
\sqrt{2}x = 3
\]

Divide both sides by \(\sqrt{2}\):

\[
x = \frac{3}{\sqrt{2}}
\]

Rationalize the denominator:

\[
x = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}
\]

### Solving the second equation:

\[
10 - \sqrt{2}x = 0
\]

Subtract 10 from both sides:

\[
-\sqrt{2}x = -10
\]

Divide both sides by \(-\sqrt{2}\):

\[
x = \frac{10}{\sqrt{2}}
\]

Rationalize the denominator:

\[
x = \frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}
\]

### Solutions:

The solutions to the equation \((\sqrt{2}x - 3)(10 - \sqrt{2}x) = 0\) are:

\[
x = \frac{3\sqrt{2}}{2} \quad \text{and} \quad x = 5\sqrt{2}
\]