Let's look at (3x)/(x^(2)+6x+8) - (2x/x+2).
This is of the general form A/B + C/D, where A = 3x, B = x^2+6x+8, C = -2x and D = x+2. You can always write this in terms of a common denominator (same bottom line) by multiplying the first term by 1 in the form of D/D and the second term also by 1 but now in the form B/B, to get AD/BD + BC/BD. Since the denominators are now the same, this is (AD+BC)/BD. In this case, you get (3x)(x+2) + (x^2+6x+8)(-2x) all over (x^2+6x+8)(x+2).
However in this case it turns out the working can be simplified somewhat. Look for a common factor in the B and the D, ie in the x^2+6x+8 and x+2 and you see that actually x+2 divides x^2+6x+8 = (x+2)(x+4). So you can leave the first term alone and multiply the second by 1 in the form (x+4)/(x+4) to get 3x/(x^2+6x+8) - 2x(x+4)/(x^2+6x+8) = (3x-2x(x+4)) / (x^2+6x+8).