Suppose a, b denote the distinct real roots of the quadratic...

Finding the sum and product of roots of the two given quadratic equations i.e.

$a + b = - 20 \text{ and } ab = - 2020 . . . . . . . . . . . \left(i\right)$

$c + d = 20 \text{ and } cd = 2020 . . . . . . . . . . . \left(ii\right)$

Now, $ac\left(a - c\right) + ad\left(a - d\right) + bc\left(b - c\right) + bd\left(b - d\right)$

$= {a}^{2}\left(c + d\right) - a\left({c}^{2} + {d}^{2}\right) + {b}^{2}\left(c + d\right) - b\left({c}^{2} + {d}^{2}\right)$

$= \left({a}^{2} + {b}^{2}\right)\left(c + d\right) - \left(a + b\right)\left({c}^{2} + {d}^{2}\right)$

$= \left\{{\left(a + b\right)}^{2} - 2ab\right\}\left(c + d\right) - \left(a + b\right)$\left\{{\left(c + d\right)}^{2} - 2cd\right\}$</p><p>$= \left\{{\left(- 20\right)}^{2} - 2\left(- 2020\right)\right\}\left(20\right) - \left(- 20\right)\left\{{\left(20\right)}^{2} - 2\left(2020\right)\right\}$(using equations \left(i\right) \text{ and } \left(ii\right)$)

$= 16000$