Option 4 : 0

**Given:**

x + (1/x) = - 1

**Formula used:**

(x - 1)(x^{2} + x + 1) = x^{3} - 1

**Calculaion:**

x + (1/x) = - 1

⇒ (x^{2} + 1)/x = - 1

⇒ x2 + 1 = - x

⇒ x2 + x + 1 = 0

Multiply the above equation by (x - 1) on both sides, we get

(x - 1)(x2 + x + 1) = (x - 1) × 0

⇒ (x - 1)(x2 + x + 1) = 0

⇒ x3 - 1 = 0

⇒ x^{3} = 1

Now, x^{9} + x^{6} - 2 = (x^{3})^{3} + (x^{3})^{2} - 2

⇒ 1^{3} + 1^{2} - 2

⇒ 1 + 1 - 2

⇒ 2 - 2

⇒ 0

**∴ The value of x ^{9} + x^{6} - 2 is 0. **

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