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

The denominators are quadratic expressions, so we need to factor them to find their common factors.

The first denominator, x^2 + 3x + 2, can be factored as (x + 1)(x + 2).
The second denominator, x^2 + 5x + 6, can be factored as (x + 2)(x + 3).
The third denominator, x^2 + 7x + 12, can be factored as (x + 3)(x + 4).
The fourth denominator, x^2 + 9x + 20, can be factored as (x + 4)(x + 5).

Now we have a common denominator of (x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

To simplify each fraction, we need to multiply the numerator and denominator of each fraction by the missing factors from the common denominator.

The first fraction, 2/(x^2 + 3x + 2), needs to be multiplied by (x + 3)(x + 4)(x + 5).
So, the first fraction becomes 2(x + 3)(x + 4)(x + 5)/(x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

The second fraction, 2/(x^2 + 5x + 6), needs to be multiplied by (x + 1)(x + 4)(x + 5).
So, the second fraction becomes 2(x + 1)(x + 4)(x + 5)/(x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

The third fraction, 2/(x^2 + 7x + 12), needs to be multiplied by (x + 1)(x + 2)(x + 5).
So, the third fraction becomes 2(x + 1)(x + 2)(x + 5)/(x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

The fourth fraction, 2/(x^2 + 9x + 20), needs to be multiplied by (x + 1)(x + 2)(x + 3).
So, the fourth fraction becomes 2(x + 1)(x + 2)(x + 3)/(x + 1)(x + 2)(x + 3)(x + 4)(x + 5).

Now we can combine the fractions by adding the numerators and keeping the common denominator:

2(x + 3)(x + 4)(x + 5) + 2(x + 1)(x + 4)(x + 5) + 2(x + 1)(x + 2)(x + 5) + 2(x + 1)(x + 2)(x + 3)
----------------------------------------------------------------------------------------------------------------
(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)

Simplifying the numerator:

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

Expanding the brackets:

= 2[(x^2 + 7x + 12)(x + 5) + (x^2 + 5x + 4)(x + 5) + (x^2 + 3x + 2)(x + 5) + (x^2 + 3x + 2)(x + 3)]

= 2[(x^3 + 12x^2 + 47x + 60) + (x^3 + 10x^2 + 25x + 20) + (x^3 + 8x^2 + 17x + 10) + (x^3 + 6x^2 + 11x + 6)]

= 2[4x^3 + 36x^2 + 100x + 96]

= 8x^3 + 72x^2 + 200x + 192

Therefore, the simplified expression is:

(8x^3 + 72x^2 + 200x + 192) / [(x + 1)(x + 2)(x + 3)(x + 4)(x + 5)]