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United States competition problems

Algebra Difficulty 4.8 AIME Prove it United States

Problem:
Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$ . What are all possible values of $x^{4}+2 x^{3}+2 x^{2}+2 x+1$ ?

Solution

Solution:
$x^{4}+2 x^{3}+2 x^{2}+2 x+1 = (x+1)\left(x^{3}+x^{2}+x+1\right) = 0$ is the only possible solution.

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