distance divided by speed equals time, so:

680/(305+w) + 810/(305-w) = 5

Get a common denominator and add:

{680(305-w) + 810(305+w)}/(305^2 - w^2) = 5

680(305) - 680w + 810(305) + 810w = 5(305^2) - 5w^2

5w^2 + 130w + 1490(305) - 5(305^2) = 0

5w^2 + 130w - 10675 = 0

Unless you have a polynomial solver on your calculator, the easiest way to solve the rest of this problem is the quadratic equation (or you could divide everything by 5 to get somewhat manageable numbers and use trial and error to get your factors or use a slide rule which provides a much easier method of trial and error than even a polynomial solver and would really impress your teacher - if they understood what a slide rule was, anyway):

x = {-b-+ sqrt(b^2 - 4ac)}/(2a)

w = {-130 -+ sqrt (130^2 - 4 * 5 * (-10675))}/10 

w = (-130 -+ 480)/10

w = (-13 -+ 48)

w = -61 or 35

When we set up the problem, we used '+' for with the wind and '-' for against the wind. While -61 is a valid answer, it will mean you went 680 miles against the wind instead of with the wind, so it's not valid for the way the problem was worded (however, it is interesting to notice that it takes the same amount of time whether you did 680 with a 35 mph wind and the 810 against a 36 mph wind takes the same amount of time as if you did 680 against a 61 mph wind and 810 with a 61 mph wind).

35 mph is the right answer.

You should plug 35 in for w in the original equation to double check since it's easy to drop a negative sign (as I originally did).

By the way, the trial and error method isn't too bad if you divide everything by 5 to get:

w^2 + 26w - 2135 = 0

Take the square root of 2135 to figure out your boundary. At least one of the terms have to be less than this (46.2). 

You also know one of your factors have to be divisible by 5.

2135/45 = 47.444 - the difference between them isn't 26

2135/35 = 61 - AND the difference between them is 26!

Then you just have to figure out which should be positive and which should be negative. Since 26 is positive, you know your factors are:

(w + 61)(w - 35) = 0

So, w+61 = 0 or w-35=0

w has to be -61 or +35