Where is the p?

An important point one should remember while solving this problem is that the degree of the remainder polynomial should be less than that of the divisor polynomial

$ax^3 + bx + c = ax(x^2 + px + 1) - apx^2 - ax + bx + c$

$\Rightarrow ax^3 + bx + c = ax(x^2 + px + 1) -ap(x^2 + px + 1) + ap^2x + ap -ax + bx + c$

$\Rightarrow ax^3 + bx + c = (ax - ap)(x^2 + px + 1) + ((ap^2 - a + b)x + ap + c)$

Here, $(ap^2 - a + b)x + ap + c$ is the remainder.

For $x^2 + px + 1$ to be a factor of $ax^3 + bx + c$ , the remainder shouldn't exist.

Hence by equating the coefficient of $x$ and the constant to 0, we get few valuable conditions.

$c + ap = 0 \Rightarrow p = \dfrac{-c}{a}$

$ap^2 - a + b = 0$

$\Rightarrow a\dfrac{c^2}{a^2} - a + b = 0 \Rightarrow c^2 - a^2 + ab = 0 \Rightarrow \boxed{a^2 - c^2 = ab}$

which is the required condition.