Need help solving a bi-quadratic polynomial.... 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.
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.
The given polynomial $x^4+3x^2+6x+10$
Can be written as

$ x^4+2x^3+2x^2-2x^3-4x^2-4x+5x^2+10x+10$

$= x^2(x^2+2x+2) - 2x(x^2+2x+2) +5(x^2+2x+2)$

$=(x^2+2x+2)(x^2-2x+5) $