How do you combine #2/(x^2+8x+15) + 1/(x^2+11x+30)#?

To combine the fractions 2/(x^2+8x+15) and 1/(x^2+11x+30), we need to find a common denominator. The denominators in this case are (x^2+8x+15) and (x^2+11x+30).

To find the common denominator, we need to factorize the denominators.

The first denominator, x^2+8x+15, can be factored as (x+3)(x+5).

The second denominator, x^2+11x+30, can be factored as (x+5)(x+6).

Now, we can see that the common denominator is (x+3)(x+5)(x+6).

To combine the fractions, we need to multiply the numerator and denominator of each fraction by the missing factors in the other denominator.

For the first fraction, we multiply the numerator and denominator by (x+6), which gives us 2(x+6)/[(x+3)(x+5)(x+6)].

For the second fraction, we multiply the numerator and denominator by (x+3), which gives us 1(x+3)/[(x+3)(x+5)(x+6)].

Now, we can add the fractions together by combining the numerators over the common denominator:

[2(x+6) + 1(x+3)]/[(x+3)(x+5)(x+6)].

Simplifying the numerator gives us (2x+12 + x+3)/[(x+3)(x+5)(x+6)].

Combining like terms in the numerator gives us (3x+15)/[(x+3)(x+5)(x+6)].

Therefore, the combined fraction is (3x+15)/[(x+3)(x+5)(x+6)].