3n^2+2 =  x^2 is impossible for all integers x.

3n^2+2 = x^2 => 3 divides x^2 -2 => x=3m,x = 3m+1 or x = 3m+2.

Case x=3m.

3n^2 = 9m^2 - 2 => 3 divides 3r+2 impossible for integers r=1,2,.. So the statement is true for squares of the form 3m.

Case x=3m+1.

 3n^2  = 9m^2+6m-1 = 3r-1 a contradiction. So, the statement is true for all squares of the form 3n+1.  

Similarly, for x = 3m+2 we have 3n^2 = 9m^2+12m+2 = 3r+2 another contridiction. So the statement is true for all squares of the form 3m+2.

Thus, we have shown the statement is true for all possible square integers and therefore true for all n. 

Cino Hilliard