algebraic fractions

I'm guessing you mean q/(q^2+5q+6) and 1/(q^2+3q+2). Because without the parenthesis it really means 1/q + 5q + 6 and 1/q^2 + 3q + 2.

So if that's correct, you can factor each of the denominators to obtain:

q / ((q+2)(q+3)) and 1 / ((q+1)(q+2)

To add these two rational expressions you need to find a common denominator. Luckily, they already have (q+2) in common. That means that the first needs (q+1) and the second needs (q+3) so you add multiply the first by (q+1)/(q+1) and the second by (q+3)/(q+3) to get the common denominator.

(q*(q+1)) / ((q+1)(q+2)(q+3)) and
(1*(q+3)) / ((q+1)(q+2)(q+3))

Now that they have a common denominator, you just add their nominators together and keep the denominator:

(q*(q+1)+1*(q+3)) / ((q+1)(q+2)(q+3))

Simplifying gives:

(q^2+q+q+3) / ((q+1)(q+2)(q+3))
(q^2+2q+3) / ((q+1)(q+2)(q+3))

Unfortunately the top polynomial is not factorable (quadratic equation returns imaginary roots). Thus, this is in simplified form (besides the fact that you could multiply out the denominator).

Hope this helps.