Need help solving a bi-quadratic polynomial....

Question

The polynomial to be factorised as a product of two factors is- $$x^4+3x^2+6x+10$$. I checked the solution in wolfram alpha to be- $$(x^2-2x+5)(x^2+2x+2)$$. I tried to factorise it by expressing it as a sum of two squares $$(x^2+1)^2+(x+3)^2$$. But I cannot solve it. Please help. Thanks a lot in advance.

Answer 1

This is going beyond my knowledge in irreducible/reducible forms.. but it seems clear that if we assume at first glance that we have (which is not the way I would go about factoring forms like this) $$ \left(x^2+ax\cdots\right)\left(x^2+bx\cdots\right) $$ then we know that $$ \left(x^2+ax\cdots\right)\left(x^2+bx\cdots\right) = x^4 +(a+b)x^3 + \cdots $$ thus we must have $a+b = 0\implies a = -b$

but like I said I know there must be some group theory approach thanks to Galois et al.