= Solution
Keep the original four-node <interpolation polynomial> $p$. The correction must vanish at all four original nodes, so a degree-at-most-four interpolant has the form
$$
q(x)=p(x)+c(x+1)x(x-1)(x-3).
$$
Since $p(2)=-1$ and the product at $2$ is $-6$, matching the new value gives $-1-6c=-7$, hence $c=1$. Expanding yields
$$
\boxed{q(x)=x^4-2x^3-3x^2+4x-3.}
$$
It takes the values $-7,-3,-3,-7,9$ at the five nodes. The same root-count argument proves uniqueness of this <interpolation polynomial>.
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