I want to determine the coefficients $w_0,w_1,w_2$ and $c$ of the integration formula $$\int_{-1}^1 f(t) \, dt = w_0f(0)+w_1(f(-1)+f(1))+w_2(f(-c)+f(c))$$ so that it integrates exactly the polynomials with the highest possible degree.
I have done similar problems to this one, for example, when finding the weighs of the Simpsons rule. However, in this case I don't know which is the highest possible order of the polynomial, I also don't know what to do with $c$.
This was a difficult problem until about $1826$ when Jacobi pointed out that for any integer $0\le k\le n$, the quadrature rule $Q_n$ has degree of exactness $n-1+k$ if and only if
(i) $Q_n$ is interpolatory
(ii) The node polynomial $\omega_n$ is orthogonal to all polynomials of degree less than $k$.
So condition (ii) gets you $c$ by orthogonality with $p(x)=x$ and then condition (i) gets you $w_0$ with $f_0(x)=(x^2-1)(x^2-c^2)$ and so on.
Hopefully the proof of the theorem will be considered obvious enough to a modern reader. Alternatively one might recognize the Gauss-Lobatto $5$-point rule in the problem statement.