I'm not quite sure how to approach this. My thinking is to use the Lagrangian to solve for the FOC and the proceed, however, I'm not sure how to proceed from there with the SOC or if that's missing the point of the question entirely.
The problem is as follow:
If the FOC for $x^2 + y^2$, s.t. $x + 2y = a$, are satisfied at a particular point, would the SOC be satisfied at that point? Why or why not?
The first-order necessary conditions are that a local minimizer must be a critical point of the Lagrangian.
The second-order sufficient condition is that the bordered Hessian of the Lagrangian must have the right alternating-sign property.
In your case, you have a convex function on a convex set, so the first-order necessary conditions are known to be sufficient. Any solution to the FONCs is a global minimizer, and since your function is also strictly convex, there must be a unique solution.