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Since x^2+px+1 is a factor of ax^3+bx+c, the other factor should be ax+c. ( by comparing x^3 term and the constant term)
So ax^3+bx+c = (x^2+px+1)(ax+c)
Comapring the x terms and x^2 terms we get
b = pc + a => 2ab = 2pca + 2a^2 …(1)
0 = c + pa. Squaring both sides we get
0 = c^2 + 2pca +a^2 ……(2)
(1) - (2) gives
2ab = 2pca + 2a^2 - c^2 - 2pca - a^2
2ab = a^2 - c^2.
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