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

Let q(x) = p(x) + 9x.

Then, by the problem:
q(1) = q(2) = q(3) = 0, and so:
q(x) is divisible by (x-1)(x-2)(x-3).

Since, the coefficient of x^4 is 1 while the coefficient of x^0 is d, it follows that:

q(x) = (x-1)(x-2)(x-3)(x- d/6); and so:
q(10) = 504(10-m)
q(-6) = 504(6+m)
Or, 1/4( q(10) + q(-6)) = 2016
Or,  1/4( p(10) + p(-6))
= 2016 - 9/4*(10-6)
= 2016 - 9
= 2007