= Solution
The <Horn copositive matrix> is invariant under cyclic permutation of its five coordinates: its negative entries correspond precisely to neighboring indices on the five-cycle. For any $x\geq0$, cyclically relabel the coordinates so that $x_5$ is a smallest coordinate. In particular, $x_4-x_5\geq0$.
Using the stated identity in this coordinate order,
$$
x^THx=(x_1-x_2+x_3-x_4+x_5)^2+4x_2x_5+4x_1(x_4-x_5)\geq0.
$$
The square is nonnegative, and both remaining terms are nonnegative since $x_i\geq0$ and $x_4\geq x_5$. Cyclic invariance means the relabeling has not changed the <quadratic form>. Thus $H$ is a <copositive matrix> for every original ordering of $x$.
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