Extend polynomial identities to complex numbers for higher-degree polynomial work.
Factor an even-power polynomial completely over the complex numbers using conjugate imaginary factors
Problem
Factor \(x^{4}~+~5x^{2}~+~4\) completely into linear factors over the complex numbers.
Big Picture
What this problem is really about
When only even powers appear, treat the polynomial as a quadratic in a lower power such as x squared. Factor that quadratic structure first, substitute the original power back, and continue until every factor is linear over the complex numbers. Any remaining sum of squares splits into an imaginary conjugate pair, and the number of linear factors should match the degree.
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