If sin θ and cos θ are the roots of the equation

Given sinθ and cosθ are roots of ax2 – bx + c = 0

Sum of roots is $\frac{b}{a}$ and product of root is $\frac{c}{a}$

i.e $\cos \theta+\sin \theta=\frac{b}{a}$ and $\cos \theta \sin \theta=\frac{c}{a}$

Since $\sin ^{2} \theta+\cos ^{2} \theta=1$

i. e $(\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta$

i.e $\left(\frac{b}{a}\right)^{2}=1+2 \frac{c}{a}$

i. e $b^{2}=a^{2}+2 a c$

i. e $a^{2}-b^{2}+2 a c$

Hence, the correct answer is option B.