To simplify the expression, we need to find a common denominator for all the fractions.

The denominators are:
x^2 + 3x + 2
x^2 + 5x + 6
x^2 + 7x + 12
x^2 + 9x + 20

To find a common denominator, we need to factorize each denominator and identify the common factors.

x^2 + 3x + 2 = (x + 1)(x + 2)
x^2 + 5x + 6 = (x + 2)(x + 3)
x^2 + 7x + 12 = (x + 3)(x + 4)
x^2 + 9x + 20 = (x + 4)(x + 5)

The common denominator is then (x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

Now, let's rewrite each fraction with the common denominator:

2/(x^2 + 3x + 2) = 2/[(x + 1)(x + 2)]
2/(x^2 + 5x + 6) = 2/[(x + 2)(x + 3)]
2/(x^2 + 7x + 12) = 2/[(x + 3)(x + 4)]
2/(x^2 + 9x + 20) = 2/[(x + 4)(x + 5)]

Now, we can add the fractions:

2/[(x + 1)(x + 2)] + 2/[(x + 2)(x + 3)] + 2/[(x + 3)(x + 4)] + 2/[(x + 4)(x + 5)]

To add fractions with different denominators, we need to find a common denominator. In this case, we already have a common denominator, so we can simply add the numerators:

(2 + 2 + 2 + 2)/[(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)]

Simplifying the numerator:

8/[(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)]

Therefore, the simplified expression is 8/[(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)].