If $\sqrt{6} : x :: \sqrt{3} : (1 + \sqrt{2})$, then x is equal to

The symbol "$:$" and "$::$" represents a proportion.

$\sqrt{6} : x :: \sqrt{3} : (1 + \sqrt{2})$ means "$\sqrt{6}$ is to $x$" as "$\sqrt{3}$ is to $(1 + \sqrt{2})$"

In equation form:

$\dfrac{\sqrt{6}}{x} = \dfrac{\sqrt{3}}{1 + \sqrt{2}}$

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Cross-multiplying:

$\sqrt{6} \times (1 + \sqrt{2}) = x \times \sqrt{3}$

$\sqrt{6} + \sqrt{6} \cdot \sqrt{2} = x\sqrt{3}$

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Simplifying $\sqrt{6} \cdot \sqrt{2}$:

$\sqrt{6} \cdot \sqrt{2} = \sqrt{12}$

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Therefore:

$\sqrt{6} + 2\sqrt{3} = x\sqrt{3}$

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Rewriting $\sqrt{6}$:

$\sqrt{6} = \sqrt{2 \times 3} = \sqrt{2} \cdot \sqrt{3}$

Substituting:

$\sqrt{2} \cdot \sqrt{3} + 2\sqrt{3} = x\sqrt{3}$

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Factoring out $\sqrt{3}$:

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

Dividing both sides by $\sqrt{3}$:

$x = \sqrt{2} + 2$

Therefore, $x = \sqrt{2} + 2$.