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Find the value of x⁴ + 4x³ + 6x² + 4x + 9, when x = -1 + i.sq root of 2

$G i v e n : x = - 1 + \sqrt{2} i \Rightarrow 4 x = - 4 + 4 \sqrt{2} i \Rightarrow x^{2} = {(- 1 + \sqrt{2} i)}^{2} = 1^{2} + {(\sqrt{2} i)}^{2} - 2 \sqrt{2} i = 1 - 2 - 2 \sqrt{2} i = - 1 - 2 \sqrt{2} i \Rightarrow 6 x^{2} = - 6 - 12 \sqrt{2} i \Rightarrow x^{4} = {(- 1 - 2 \sqrt{2} i)}^{2} = 1^{2} + {(2 \sqrt{2} i)}^{2} + 2 \times 2 \sqrt{2} i = 1 - 9 + 4 \sqrt{2} i = - 8 + 4 \sqrt{2} i \Rightarrow x^{3} = {(- 1)}^{3} + {(\sqrt{2} i)}^{3} + 3 \sqrt{2} i - 3 \times {(\sqrt{2} i)}^{2} = - 1 - 2 \sqrt{2} i + 3 \sqrt{2} i + 6 = 5 + \sqrt{2} i \Rightarrow 4 x^{3} = 20 + 4 \sqrt{2} i \Rightarrow x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 9 = - 8 + 4 \sqrt{2} i + 20 + 4 \sqrt{2} i - 6 - 12 \sqrt{2} i - 4 + 4 \sqrt{2} i + 9 = 11$

-14

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