To simplify the expression, we can combine the fractions by finding a common denominator.

The common denominator for all the fractions is (x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56).

Now, let's add the fractions:

A = (1/(x^2 + 9x + 20)) + (1/(x^2 + 11x + 30)) + (1/(x^2 + 13x + 42)) + (1/(x^2 + 15x + 56))

To add the fractions, we need to find a common denominator. Since the denominators are already factored, the common denominator is simply the product of the denominators:

A = (1/(x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56)) * ((x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56) + (x^2 + 9x + 20)(x^2 + 13x + 42)(x^2 + 15x + 56) + (x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 15x + 56) + (x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42))

Now, we can simplify the expression by multiplying out the numerators:

A = (1/(x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56)) * (x^8 + 50x^6 + 1001x^4 + 10010x^2 + 30030 + x^8 + 48x^6 + 858x^4 + 6860x^2 + 18480 + x^8 + 46x^6 + 715x^4 + 5985x^2 + 16632 + x^8 + 44x^6 + 572x^4 + 4840x^2 + 13440)

Now, we can combine like terms:

A = (1/(x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56)) * (4x^8 + 188x^6 + 3146x^4 + 28275x^2 + 78582)

Therefore, the simplified expression is A = (4x^8 + 188x^6 + 3146x^4 + 28275x^2 + 78582)/(x^2 + 9x + 20)(x^2 + 11x + 30)(x^2 + 13x + 42)(x^2 + 15x + 56).