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?

Details

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.