To solve this equation, we need to expand the left side of the equation and then set it equal to 840.
Expanding the left side:
(x-1)(x-2)(x-3)(x-4)= (x^2 - 2x - x + 2)(x^2 - 3x - 2x + 6)(x^2 - 4x - 3x + 12)= (x^2 - 3x + 2)(x^2 - 5x + 6)(x^2 - 7x + 12)= (x^4 - 5x^3 + 6x^2 - 3x^3 + 15x^2 - 18x + 2x^2 - 10x + 12)(x^2 - 7x + 12)= (x^4 - 8x^3 + 23x^2 - 28x + 12)(x^2 - 7x + 12)= x^6 - 7x^5 + 12x^4 - 8x^5 + 56x^4 - 96x^3 + 23x^4 - 161x^3 + 276x^2 - 28x^3 + 196x^2 - 336x + 12x^2 - 84x + 144= x^6 - 15x^5 + 91x^4 - 257x^3 + 368x^2 - 420x + 144
Now setting this equal to 840:
x^6 - 15x^5 + 91x^4 - 257x^3 + 368x^2 - 420x + 144 = 840
This is a sixth-degree polynomial equation that can be solved using numerical methods or a graphing calculator.
To solve this equation, we need to expand the left side of the equation and then set it equal to 840.
Expanding the left side:
(x-1)(x-2)(x-3)(x-4)
= (x^2 - 2x - x + 2)(x^2 - 3x - 2x + 6)(x^2 - 4x - 3x + 12)
= (x^2 - 3x + 2)(x^2 - 5x + 6)(x^2 - 7x + 12)
= (x^4 - 5x^3 + 6x^2 - 3x^3 + 15x^2 - 18x + 2x^2 - 10x + 12)(x^2 - 7x + 12)
= (x^4 - 8x^3 + 23x^2 - 28x + 12)(x^2 - 7x + 12)
= x^6 - 7x^5 + 12x^4 - 8x^5 + 56x^4 - 96x^3 + 23x^4 - 161x^3 + 276x^2 - 28x^3 + 196x^2 - 336x + 12x^2 - 84x + 144
= x^6 - 15x^5 + 91x^4 - 257x^3 + 368x^2 - 420x + 144
Now setting this equal to 840:
x^6 - 15x^5 + 91x^4 - 257x^3 + 368x^2 - 420x + 144 = 840
This is a sixth-degree polynomial equation that can be solved using numerical methods or a graphing calculator.