 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 formSince $p(2)=-1$ and the product at $2$ is $-6$, matching the new value gives $-1-6c=-7$, hence $c=1$. Expanding yieldsIt takes the values $-7,-3,-3,-7,9$ at the five nodes. The same root-count argument proves uniqueness of this interpolation polynomial.