Answer: y = x^3+x^2+4x+4
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Explanation:
Assuming the coefficients are real numbers, this means 2i pairs up with -2i. They are conjugate pair roots. So we have these three roots
-1, 2i, -2i
Meaning that
x = -1, x = 2i, x = -2i
x+1 = 0, x^2 = -4 .... see note below
x+1 = 0 or x^2+4 = 0
(x+1)(x^2+4) = 0
x(x^2+4)+1(x^2+4) = 0
x^3+4x+x^2+4 = 0
x^3+x^2+4x+4 = 0
To check the answer, plug each root (-1, 2i and -2i) into this equation. You should get 0 on the left side each time.
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note: If we have x = 2i, then we can square both sides to get x^2 = 4i^2 which becomes x^2 = -4. The same happens to x = -2i. You should find that solving x^2 = -4 leads back to the two roots x = 2i or x = -2i.
