What I understand is you are trying to square these terms and arrive at an equation which is of the nature x^4, which leads you to believe that it has 4 roots. However, that is not always the case, any equation of the nature x^n will have n roots, but for these to be real it would have to satisfy some condition. If it does not then the roots are imaginary. That applies to the expression you get. However I am not sure what that condition is for an expression of the 4th order.
Hope this helps.
btw, the explaination given for the above problem is
This equation has no roots. sqrt(x^2 + 1) is bigger than or equal to 1, sqrt(x^2 + 2) is bigger than or equal to sqrt(2). Therefore, sqrt(x^2 + 1) + sqrt(x^2 + 2) is bigger than or equal to 1 + sqrt(2), which is bigger than 2.
I can't seem to understand this logic.