A problem by Diego E. Nazario Ojeda
A member of Brilliant remembers the previous system, in which members received \(4\) weekly problems of each topic, with \(3\) opportunities for each problem. He remembers that one week, in Number Theory, he got all the problems correct, but he gave two wrong answers previously. This event is possible in \(x\) ways. If \(-x\) is the only root of a polynomial of degree \(2\), what is the sum of the coefficients of that polynomial?
The order of the answers doesn't matters. Getting a wrong answer in the second and third problem is the same if you answer first the second, or the third.