Given the quadratic polynomial $ax^2 + bx + c$, find a new polynomial with coefficients expressed in terms $a$, $b$ and $c$ such that the product and the sum of its zeros will be the sum and the product, respectively, of the zeros of the original polynomial.
Recall that for the quadratic polynomial $ax^2 + bx + c$, the sum of the roots is $-\dfrac{b}a$ while the product of the roots is $\dfrac{c}a$. Now you want a quadratic whose sum is $\dfrac{c}a$ and the product is $-\dfrac{b}a$. Hence the quadratic you are after is $$\alpha \left(y^2 - \dfrac{c}ay - \dfrac{b}a\right)$$ where $\alpha$ is some constant. Since you want one such polynomial, we can choose $\alpha = a$, and this gives us $$ay^2 - cy - b$$