Amazing and beautiful question indeed!
The equation is indeed jarring and scary, and is tempting to try to solve by squaring or taking roots (I know I did, tried for almost 3 minutes

)
But after reading again, I find a 30 seconds solution by realizing that
this is not an algebra but a number property/theory problemreiterating the question
\(\sqrt{x^2 + 1} + \sqrt{x^2 + 2} = 2\)
Let's think for a sec. Okay, so we need two roots of a number that equals 2. Fair enough.
Also notice that both of the terms have an \(x^2\) term, which is always positive or zero, \(x>=0\). Up to this point, they're not helping at all.
But wait! Each of the terms are always greater than 1!
For the first term \(\sqrt{x^2 + 1} \), min value is 1 when \(x^2 = 0\)
For the second term \(\sqrt{x^2 + 2} \), min value is \(\sqrt{2} \approx 1.4 \) when \(x^2 = 0\).
You dont actually need the approximate number of 1.4 to solve this problem, only that obviously \(\sqrt{2} > \sqrt{1} \) or that \(\sqrt{2}\) MUST BE greater than 1
Now, that since the first square root term is 1 and the second is greater than 1, adding both CAN NOT BE EQUAL to 2, thus it doesn't have any solution in real numbers, hence A