2x2+3x+6 has only complex zeroes.
This can be written:
2(x2+3x/2+3)=
2(x2+3x/2+9/16-9/16+3)=
2((x+¾)2+39/16)=2((x+¾)2-(-√39/4)2).
This is now presented fundamentally as difference of two squares, which can be factorised:
2(x+¾-√(-39)/4)(x+¾+√(-39)/4). Since there is no real square root of a negative number, this can't be factorised into real factors.