<s>[INST] Let $a$, $b$, $c$, $d$ be the roots of $x^4 + x + 1 = 0$. Find $a^4 + b^4 + c^4 + d^4$. 
Let $p(x) = x^4 + x + 1 = 0$, and let $a$, $b$, $c$, $d$ be its roots.
  Find $a^4 + b^4 + c^4 + d^4$.

I have no idea how to start solving this problem.
 [/INST] For any of the roots, $r^4=-r-1$ so that the requested sum is $-a-b-c-d-4$, and by the Vieta's formula, the sum of the roots is $0$.
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