To solve this equation, we can first expand the left side and then set it equal to 5760:

(x+4)(x+6)(x+8)(x+10) = 5760
(x^2 + 6x + 4x + 24)(x^2 + 10x + 8x + 80) = 5760
(x^2 + 10x + 24)(x^2 + 18x + 80) = 5760
x^4 + 18x^3 + 80x^2 + 10x^3 + 180x^2 + 800x + 24x^2 + 432x + 1920 = 5760
x^4 + 28x^3 + 284x^2 + 1232x + 1920 = 5760

Now we can set the equation equal to zero:

x^4 + 28x^3 + 284x^2 + 1232x + 1920 - 5760 = 0
x^4 + 28x^3 + 284x^2 + 1232x - 3840 = 0

Unfortunately, the equation is quite complex and may be difficult to solve algebraically. However, it can be solved using numerical methods such as graphing or using a computer algebra system.