To solve this equation, we can start by getting rid of the denominators. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 2010 * 2002 * 2001.

So, we have:

(2010 * 2002 * 2001) * [(x + 1)/2012 + (x + 2)/2011 + (x + 3)/2010] = (2010 * 2002 * 2001) * [(x + 10)/2003 + (x + 11)/2002 + (x + 12)/2001]

Simplifying both sides, we get:

(2002 * 2001 * (x + 1)) + (2010 * 2001 * (x + 2)) + (2010 * 2002 * (x + 3)) = (2012 * 2002 * (x + 10)) + (2012 * 2001 * (x + 11)) + (2011 * 2003 * (x + 12))

Expanding both sides, we get:

(4004002x + 4004002) + (4022010x + 8044020) + (4024020x + 12072060) = (4024024x + 40240240) + (4024024x + 44264264) + (4026033x + 48372396)

Combining like terms, we get:

12036032x + 24144082 = 12036081x + 132676900

Subtracting 12036032x from both sides, we get:

24144082 = 12036049x + 132676900

Subtracting 132676900 from both sides, we get:

-108532818 = 12036049x

Dividing both sides by 12036049, we get:

x = -108532818/12036049

Simplifying the fraction, we get:

x = -9

Therefore, the solution to the equation is x = -9.