Express polynomial as a product of real quadratic polynomials with no real roots

Question

I want to express

$$P(x) = x^4 + x^2 + 1$$

as as a product of real quadratic polynomials with no real roots.

I know that: $$(x^2 - bx + a^2)(x^2 +bx + a^2)$$ $$ =x^2 + (2a^2 - b^2)x^2 + a^4 $$

I thought that $b^2 = 2a^2 - 1$ in this case and provided a = 1, b = $\pm 1$. However in my textbook, the answer given is $b^2 = 2a^2 + 1 \therefore b = \sqrt{3}$.

I'm not sure where I've went wrong here.

Answer 1

Note that

$$x^4+x^2+1 = (x^2-x+1)(x^2+x+1)$$

and

$$x^4-x^2+1 = (x^2-\sqrt{3}x+1)(x^2+\sqrt{3}x+1)$$

Considering the answer given by your textbook, it seems that your textbook wanted you to factorize $x^4-x^2+1$ into quadratic factors. If you haven't misread the problem, it's worth notifying the authors of this error/typo in their book.