= Solution
Only the symmetric part of $M$ contributes to $x^TMx$. The functions
$$
x\longmapsto x^TMx
$$
therefore form a real vector space of dimension
$$
D=\frac{d(d+1)}2=\binom{d+1}{2},
$$
with coordinates given by the quadratic monomials $x_i^2$ and $x_ix_j$ for $i<j$. The theorem on signs of a finite-dimensional function space says that the sign class of a $D$-dimensional real vector space has VC dimension at most $D$. Consequently
$$
\boxed{VC(H_3)\leq\binom{d+1}{2}.}
$$
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