We can write p(x)=(x-a)(x-1)(x-2)(x-3) to include the roots and the coefficient of x^4 equalling 1. The fourth root is a.
p(0)=-a(-1)(-2)(-3)=6a (the constant term); p(6)=(6-a)(5)(4)(3)=360-60a.
p(0)+p(6)=360-60a+6a=360-54a.
We are told all the roots, assumedly, so there must be a duplicated root and a=1, 2 or 3. So p(0)=6, 12 or 18.
Therefore, p(0)+p(6)=36, -288 or -612.