To solve this equation, we first need to expand the left side:

(x+1)(x+4)(x+2)(x+3) = (x^2 + 5x + 4)(x^2 + 5x + 6)
= x^2(x^2 + 5x + 6) + 5x(x^2 + 5x + 6) + 4(x^2 + 5x + 6)
= x^4 + 5x^3 + 6x^2 + 5x^3 + 25x^2 + 30x + 4x^2 + 20x + 24
= x^4 + 10x^3 + 35x^2 + 50x + 24

So the equation becomes:

x^4 + 10x^3 + 35x^2 + 50x + 24 = 5040

Now, set the equation equal to zero:

x^4 + 10x^3 + 35x^2 + 50x + 24 - 5040 = 0

This simplifies to:

x^4 + 10x^3 + 35x^2 + 50x - 5016 = 0

Unfortunately, this equation is not easily solvable by factoring or other simple methods. You may need to use numerical methods or a graphing calculator to find the roots of this polynomial equation.