The HCF of x6 - 1 and x4 + 2x3 - 2x1 - 1 is = ?

Answer: Option C

$$\begin{array}{rl}& x^6-1\Rightarrow {\left(x^2\right)}^3-1^3\\ & \text{Using}\\ & \Rightarrow a^3-b^3=\left(a-b\right)\left(a^2+b^2+ab\right)\\ & \Rightarrow \left(x^2-1\right)\left(x^4+1+x^2\times 1\right)\\ & \Rightarrow \left(x^2-1\right)\left(x^4+1+x^2\right).....(i)\\ & \text{Again}\\ & x^4+2x^3-2x-1\\ & \Rightarrow x^4-1+2x\left(x^2-1\right)\\ & \Rightarrow {\left(x^2\right)}^2-1^2+2x\left(x^2-1\right)\\ & \Rightarrow \left(x^2-1\right)\left(x^2+1\right)+2x\left(x^2-1\right)\\ & \Rightarrow \left(x^2-1\right)\left(x^2+1+2x\right).....(ii)\end{array}$$
  ⇒ From equation (i) and (ii) HCF is a common term HCF = (x² - 1)