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

$3+\sqrt{3}$

The correct answer is Option (1) → $3+\sqrt{3}$

Interpret the proportion as involving $\sqrt{3}$:

$(2-\sqrt{3}) : (1+\sqrt{3}) :: (6-3\sqrt{3}) : x$

Notice that

$6-3\sqrt{3} = 3(2-\sqrt{3})$

So the second ratio is obtained by multiplying the first term by 3.
Hence, the corresponding second term must also be multiplied by 3:

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