The problem asks to multiply and simplify the expression $(\sqrt{2} + 3\sqrt{x})(4\sqrt{2} + 6\sqrt{x})$, assuming that $x$ represents a non-negative number.

1. Problem Description

The problem asks to multiply and simplify the expression $(\sqrt{2} + 3\sqrt{x})(4\sqrt{2} + 6\sqrt{x})$, assuming that $x$ represents a non-negative number.

2. Solution Steps

To multiply the given expression, we use the distributive property (also known as the FOIL method):

$(\sqrt{2} + 3\sqrt{x})(4\sqrt{2} + 6\sqrt{x}) = \sqrt{2}(4\sqrt{2}) + \sqrt{2}(6\sqrt{x}) + 3\sqrt{x}(4\sqrt{2}) + 3\sqrt{x}(6\sqrt{x})$

$= 4(\sqrt{2})^2 + 6\sqrt{2x} + 12\sqrt{2x} + 18(\sqrt{x})^2$

$= 4(2) + 6\sqrt{2x} + 12\sqrt{2x} + 18x$

$= 8 + 18\sqrt{2x} + 18x$

We can rearrange the terms to write the expression as:

$18x + 18\sqrt{2x} + 8$

3. Final Answer

$18x + 18\sqrt{2x} + 8$