Let p(x)=x^4+ax^3+bx^2+cx+d, where a, b, c, and d are constants.

If p(1)=-9, p(2)=-18, and p(3)=-27, find the value of
¼(p(10)+p(-6)).

...then it's -9. Taking p(x)=-9x + a fourth degree polynomial with the roots 1, 2, 3 (and another unknown one) fits the given points, so

if the solution is unique, it must be the same as for this p(x).