To solve this equation, we first need to find a common denominator for the fractions on the left side.

The common denominator for the fractions 1/x(x+2) and 1/(x+1)^2 is x(x+2)(x+1)^2.

So, the equation becomes:

(x+1)^2/(x(x+2)(x+1)^2) - x(x+2)/(x(x+2)(x+1)^2) = 1/12

Expanding the fractions, we get:

(x^2 + 2x + 1 - x^2 - 2x)/(x(x+2)(x+1)^2) = 1/12

1/(x(x+2)(x+1)^2) = 1/12

Now, cross multiply to solve for x(x+2)(x+1)^2:

12 = x(x+2)(x+1)^2

Now, expand and simplify the right side:

12 = x(x^3 + 3x^2 + 3x + 2)

12 = x^4 + 3x^3 + 3x^2 + 2x

Rearrange the equation to form a polynomial:

x^4 + 3x^3 + 3x^2 + 2x - 12 = 0

This quartic equation can be solved by factoring, using the rational root theorem, or by numerical methods.