Suppose $a$, $b$ denote the distinct real roots of the quadratic polynomial $x^{2}+20x-2020$ and suppose $c$, $d$ denote the distinct complex roots of the quadratic polynomial $x^{2}-20x+2020$. Then the value of $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d)$ is

$a,\ b \text{ are the distinct real roots of } x^2 + 20x - 2020$ $c,\ d \text{ are the distinct complex roots of } x^2 - 20x + 2020$ $\text{To find: } ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)$ $(1)\ 0 \qquad (2)\ 8000 \qquad (3)\ 8080 \qquad (4)\ 16000$ $1.\ \text{roots} \to \text{sum and product (Vieta)}$ $2.\ \text{group and simplify the expression}$ $3.\ \text{substitute and evaluate}$ $a + b = -20, \qquad ab = -2020$ $c + d = 20, \qquad cd = 2020$ $c^2 + d^2 = (c+d)^2 - 2cd = 400 - 4040 = -3640$ $E = a\big[a(c+d) - (c^2+d^2)\big] + b\big[b(c+d) - (c^2+d^2)\big]$ $E = a(20a + 3640) + b(20b + 3640)$ $E = 20(a^2+b^2) + 3640(a+b)$ $a^2 + b^2 = (a+b)^2 - 2ab = 400 + 4040 = 4440$ $E = 20(4440) + 3640(-20)$ $E = 88800 - 72800 = 16000$ $\therefore\ \text{value} = 16000 \Rightarrow \boxed{\text{option (4)}}$