= Solution
A nonempty topological space is irreducible when it is not the union of two proper closed subsets, equivalently when any two nonempty open subsets intersect.
The Zariski topology on $\mathbb A^n$ is Noetherian because $k[x_1,\ldots,x_n]$ satisfies the ascending chain condition on <ideals>. Suppose a closed set $X$ that is not a finite union of irreducible closed sets were minimal among such counterexamples. It is reducible, say $X=X_1\cup X_2$ with both $X_i$ proper closed subsets. Minimality expresses each $X_i$ as a finite union of irreducible closed sets, and combining those decompositions contradicts the choice of $X$. Thus every closed $X$ has a finite irreducible decomposition.
Using $x_3=x_2^2$, the second defining <polynomial> reduces to
$$
x_1^2-x_2^2-x_2^4+x_3^2=x_1^2-x_2^2.
$$
Consequently
$$
Z(x_3-x_2^2,x_1^2-x_2^2-x_2^4+x_3^2)
=
Z(x_3-x_2^2,x_1-x_2)
\cup
Z(x_3-x_2^2,x_1+x_2).
$$
Each component is the image of $t\mapsto(\pm t,t,t^2)$ and is isomorphic to $\mathbb A^1$, hence irreducible. In characteristic two the two displayed components coincide.
Suppose a <polynomial> $P(x,y)=\sum_{j=0}^m p_j(x)y^j$ vanishes on every $(x,e^x)$. Then
$$
\sum_{j=0}^m p_j(x)e^{jx}=0
$$
as an <entire function>. These exponential-polynomial <functions> are linearly independent: applying $(D-m)^{\deg p_m+1}$ kills the term with largest exponent, while it acts injectively on $p_j(x)e^{jx}$ for $j<m$; induction on the number of terms finishes the proof. Hence every $p_j$ is zero, so no nonzero <polynomial> vanishes on the graph. Its Zariski closure is all of $\mathbb A^2$.
Finally refine an open cover of an affine variety $X$ by distinguished opens $D(f_j)$, choosing one inside an original member around each point. Since these distinguished opens cover,
$$
V((f_j)_j)=\varnothing.
$$
The <Hilbert Nullstellensatz> implies that the $f_j$ generate the unit <ideal>. Thus
$$
1=g_1f_{j_1}+\cdots+g_rf_{j_r}
$$
for finitely many of them, so $D(f_{j_1}),\ldots,D(f_{j_r})$ cover $X$. The corresponding original <open sets> form a finite subcover. This is <quasi-compactness of an affine variety>.
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