[syzar]

It leads me to think about writing a polynomial function that I know will have complex roots because I was taught the answer to that is "no solution",

By the time one reaches calculus they have been taught that complex solutions are valid solutions. They just aren’t real solutions. Therein lies one of the problems teachers of mathematics have. Conveying the concept of the answer depending on what the current algebraic object one is working on. We have to hand wave do some brain washing because the nuances involved are far too complicated for the students to understand at this level.