= Solution
Let $p(x)=\prod_{i=0}^n(x-x_i)$ be the monic node <polynomial>. Suppose first that the quadrature is exact through degree $2n+1$. For any <polynomial> $r$ of degree at most $n$, $pr$ has degree at most $2n+1$, while every sampled value is zero. Exactness gives
$$
\int_a^b p(x)r(x)\,dx=\sum_i a_i p(x_i)r(x_i)=0.
$$
Thus $p$ is an <orthogonal polynomial> of degree $n+1$, proving necessity in the <orthogonal-node criterion for Gaussian quadrature>.
Conversely, suppose the nodes are the zeros of such a <polynomial>. Rescale it to be monic without changing its zeros or orthogonality. For any $f$ of degree at most $2n+1$, <polynomial> division gives
$$
f=pq+r,\qquad\deg q\le n,\quad\deg r\le n.
$$
Orthogonality makes $\int pq=0$; the quadrature of $pq$ is zero because $p(x_i)=0$. The assumed degree-$n$ exactness applies to $r$, so
$$
\int_a^b f=\int_a^b r=\sum_i a_i r(x_i)=\sum_i a_i f(x_i).
$$
Hence \b[degree-$2n+1$ exactness is equivalent to orthogonal nodes]. The given distinct-zero property ensures the $n+1$ node values are independent interpolation data; the degree-$n$ exactness fixes the corresponding interpolatory weights. This is the construction of <Gaussian quadrature>.
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