Value of Symmetric Polynomial Expression in Quadratic Roots

To find the value of the given expression, let the expression be denoted by $E$: $E = ac(a - c) + ad(a - d) + bc(b - c) + bd(b - d)$

Step 1: Use Vieta's formulas for both quadratic polynomials

For the quadratic equation $x^2 + 20x - 2020 = 0$, the roots are $a$ and $b$. By Vieta's formulas: $a + b = -20$ $ab = -2020$

For the quadratic equation $x^2 - 20x + 2020 = 0$, the roots are $c$ and $d$. By Vieta's formulas: $c + d = 20$ $cd = 2020$

Step 2: Simplify the expression $E$

Expand the given expression: $E = a^2c - ac^2 + a^2d - ad^2 + b^2c - bc^2 + b^2d - bd^2$

Grouping the terms with $a^2, b^2$ together and $c^2, d^2$ together: $E = a^2(c + d) + b^2(c + d) - c^2(a + b) - d^2(a + b)$ $E = (a^2 + b^2)(c + d) - (c^2 + d^2)(a + b)$

Step 3: Calculate $a^2 + b^2$ and $c^2 + d^2$

Using the algebraic identity $x^2 + y^2 = (x + y)^2 - 2xy$:

1. For $a^2 + b^2$: $a^2 + b^2 = (a + b)^2 - 2ab$ $a^2 + b^2 = (-20)^2 - 2(-2020) = 400 + 4040 = 4440$

2. For $c^2 + d^2$: $c^2 + d^2 = (c + d)^2 - 2cd$ $c^2 + d^2 = (20)^2 - 2(2020) = 400 - 4040 = -3640$

Step 4: Substitute the values into the expression $E$

$E = (4440)(20) - (-3640)(-20)$ $E = 88800 - 72800 = 16000$

Thus, the value of the expression is $16000$.

Correct Option: (D)