If $x^2 +px +1$ is a factor of $ ax^3 +bx+c$ then relate $a,b,c$ Suppose If $x^2 +px +1$ is a factor of $ax^3 +bx+c$ then relate $a,b,c$ such that $a,b,c \in R$
I can write $$ax^3 +bx+c=(x^2 +px +1)(\lambda x +D)$$
$$\implies ax^3 +bx+c =\lambda x^3 + x^2.p\lambda + x(\lambda+pD)+D $$
and then compare coefficient to find out relation but that will be long and tedious process , I want shorter approach to this problem . Btw I was given following options for this question

A) $a^2+c^2+ab=0$
B)  $a^2-c^2+ab=0$
C) $a^2-c^2-ab=0$
D) $ap^2+bp+c=0$

Maybe we can relate something by looking at options?

Just observe that the product of two roots of the quadratic is $1$. So the third root has to be $\frac{-c}{a}$. Now substitute this root instead of $x$ in the cubic.
