All ordered pairs $(a, b)$ where $a$, $b$ coefficients of quadratic

Question

Suppose the quadratic $x^2 + ax + b$ equals $0$ when $x = c$ or $x = d$. If $c^2d + d^2c = 10$, and $a$ and $b$ are integers, what are all possible ordered pairs $(a, b)$?

Answer 1

Sum of roots $= - \dfrac{\text{coeff} (x)}{\text{coeff}(x^2)} = -a \Rightarrow \color{blue}{c+d = -a}$
Product of roots $= \dfrac{\text{const. term}}{\text{coeff}(x^2)} = b \Rightarrow \color{blue}{cd = b}$

$c^2d+d^2c = cd(c+d) = -ab$

Could you proceed further?