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

(x+2014)(x+2015)(x+2016) = x^3 + 6035x^2 + 6072011x + 4076204

Now we set this expression equal to 24:

x^3 + 6035x^2 + 6072011x + 4076204 = 24

Next, we subtract 24 from both sides to set the equation to zero:

x^3 + 6035x^2 + 6072011x + 4076180 = 0

Now we can look for the roots of this equation. One possible root can be found by trial and error; plugging in small integers as potential solutions.

By trying x = -2, we find that it is a root of the equation. Therefore, we can factor out (x+2) and perform polynomial division to find the other roots:

(x+2)(x^2 + 6033x + 2038090) = 0

Setting x^2 + 6033x + 2038090 = 0, we can use the quadratic formula:

x = (-6033 ± sqrt(6033^2 - 412038090)) / 2

x = (-6033 ± sqrt(36472489 - 8152360)) / 2

x = (-6033 ± sqrt(28320129)) / 2

x = (-6033 ± 5320.40) / 2

Therefore, the roots of the equation are -2, -3006.2, and -3026.8.