# Common root of cubic and quadratic equation

If equations $$ax^3+2bx^2+3cx+4d=0$$ and $$ax^2+bx+c=0$$ have a non zero common root, prove that $$(c^2-2bd)(b^2-ac) \geq 0$$.

I know the condition of common root of two quadratic equations but I have no idea on how to proceed with this question.

You can multiply the second by $$x$$ and subtract from the first. That leaves $$(2b-a)x^2+(3c-b)x+(4d-c)=0$$ and $$ax^2+bx+c=0$$ as two quadratic equations, which you know how to handle.