Give the complete factorization and roots of \((x~+~1)(x^{2}~+~4)\) over the real numbers and over the complex numbers.

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Over R and C: (x+1)(x-2)(x+2). Roots: -1, 2, and -2.

Over R: (x+1)(x²+4). Over C: (x+1)(x-2i)(x+2i). Roots: -1, 2i, and -2i.

Over R: (x+1)(x²+4). Over C: (x+1)(x-2i), omitting root -2i.

Over R: (x-1)(x²+4). Over C: (x-1)(x-2i)(x+2i). Roots: 1, 2i, and -2i.

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Curriculum context

Course: Math III
Standard: N-CN.9
Category: Number and Quantity
Domain: The Complex Number System
Objective: Know the Fundamental Theorem of Algebra and connect it to polynomial roots.
Problem type: Relate concrete complete factorizations over R and C