Know the Fundamental Theorem of Algebra and connect it to polynomial roots.
Enumerate polynomial root structures by degree slots
Problem
A \(\text{degree}-4\) real-coefficient polynomial has \(\text{two}~\text{specified}~\text{simple}~\text{real}~\text{roots}\). Describe every possible structure for the \(\text{remaining}~\text{two}~\text{degree}~\text{slots}\), counting multiplicity.
Big Picture
What this problem is really about
Treat the degree as a fixed number of multiplicity slots. Account for the stated simple roots first, then partition the remaining slots among additional real-root occurrences or nonreal roots. For real coefficients, nonreal slots must be grouped into conjugate pairs, while real slots may represent distinct roots or repeated occurrences of the same root.
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