= Solution
Writing $g=e^{x^{\mathsf T}x}$, the <gradient> and <Hessian matrix> are
$$
\nabla g(x)=2gx,\qquad H(g)(x)=g(2I+4xx^{\mathsf T}).
$$
Consequently, for any nonzero $v$,
$$
v^{\mathsf T}H(g)(x)v=g\left(2\|v\|^2+4(x^{\mathsf T}v)^2\right)>0.
$$
In particular the <Hessian matrix> is a <positive semidefinite matrix> everywhere, so the <Hessian criterion for convexity> proves \b[$g$ is <convex>]. The strict inequality along every nonconstant line also proves <strict convexity>.
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