1.8 Quadratic factor $q(x)=x^{2}+2bx+c.$

(i) If $b^{2}>c$, then there are distinct real roots

$$\alpha=-b+\sqrt{b^{2}-c},\qquad\beta=-b-\sqrt{b^{2}-c}$$

and the quadratic reduces to a product of real linear factors

$$x^{2}+2bx+c=(x-\alpha)(x-\beta).$$

(ii) If $b^{2}=c$, then there is a double real root

$$\alpha=\beta=-b$$

and the quadratic factors reduces to

$$x^{2}+2bx+c=(x-\beta)^{2}.$$

(iii) If $b^{2}<c$, then the quadratic has no real roots and is irreducible.