Never Judge a Problem by its Title

Algebra Level 5

Let $\displaystyle P$ and $\displaystyle Q$ be two quadratic polynomials, such that $\displaystyle P(z)=3z^2+4z+3$ and $\displaystyle Q(z)=3z^2+6z+15$ .

Furthermore, the product of $\displaystyle P(x)$ and $\displaystyle Q(y)$ for some $\displaystyle x,y \in \mathbb{R}$ is $\displaystyle 20$ . Find their sum, ie, $\displaystyle P(x)+Q(y)$ .

It can be expressed as $\displaystyle \dfrac{a}{b}$ for coprime positive integers $\displaystyle a$ and $\displaystyle b$ . Find $\displaystyle a+b$ .