= Solution
For a self-adjoint <Hamiltonian operator> bounded below, the <Rayleigh-Ritz variational principle> says $E[\psi]=\langle\psi,H\psi\rangle/\langle\psi,\psi\rangle\ge E_0$. Expanding in energy <eigenstates> proves this, since the quotient is a weighted average of energies. Optimize a tractable trial family to obtain an upper bound on the ground energy.
If $H\psi_0=E_0\psi_0$, a regular trial error $\psi=\psi_0+\epsilon\eta$ gives the exact cancellation
$$
E[\psi]-E_0=\frac{\epsilon^2\langle\eta,(H-E_0)\eta\rangle}{\|\psi_0+\epsilon\eta\|^2}=O(\epsilon^2).
$$
The linear terms vanish because $(H-E_0)\psi_0=0$. Here $\eta$ is a fixed finite-energy direction, or is uniformly bounded in the quadratic-form norm. A Hilbert-space norm bound alone is insufficient for an unbounded <Hamiltonian operator>: a norm-$\epsilon$ admixture of an <eigenstate> with energy of order $\epsilon^{-2}$ can have order-one energy error. The usual quadratic-error claim therefore presupposes a controlled-energy perturbation.
For the first excited energy minimize the quotient on states orthogonal to the exact <ground state>. Alternatively diagonalize $H$ in a trial subspace and use the min-max principle: its second Ritz <eigenvalue> is an upper bound for the second exact <eigenvalue>. An arbitrary trial function without either <orthogonality> or a min-max construction need not bound the first excited level.
The one-dimensional even potential has even <ground state> and odd <first excited state>, by the Sturm oscillation ordering. Choose the automatically orthogonal odd trial function $\psi_a(x)=Nx e^{-ax^2/2}$, $a>0$. <Integration by parts> gives the kinetic numerator $\int|\psi_a'|^2$, and the given Gaussian moments give
$$
E_1\le E(a)=\frac{3a}{2}+\frac3{2a}+\frac{15}{4a^2}.
$$
For example, the kinetic factor is $(I_0-2aI_2+a^2I_4)/I_2=3a/2$, while the potential factors are $I_4/I_2$ and $I_6/I_2$. Differentiation gives the unique positive minimizer $a^3-a-5=0$. Thus
$$
\boxed{a\approx1.90416,\qquad E_1\le E(a)\approx4.67824.}
$$
This is an upper estimate, not an exact <eigenvalue>. Improve it with odd <polynomial>-Gaussian trial functions such as $(x+bx^3)e^{-ax^2/2}$, or a larger odd Ritz basis, and optimize the width. The <parity> restriction keeps all such states orthogonal to the even <ground state>.
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