As usual, we need common denominators to add fractions. To find the Lowest Common Denominator (LCD), we need to see the denominators in factored form:
If you have trouble looking at this and seeing what the LCD is, then I find it helpful to write the two factored denominators in the following way:
6x-30 = 2 * 3 * (x-5)
4x-40 = 2 * (x-5) * 2
LCD = 2 * 3 * (x-5) * 2
Notice how I rearranged the order and the spacing on the second line so that only common factors were in the same column as factors in the first line. Once the factors are lined up like this, the LCD is simply the product of all the columns!
Now that we know what the LCD is, we look at each fraction's denominator and figure what factors of the LCD is it missing, if any? We can use the table above for this, too. The first line has a "hole" at the end where the second factor of 2 is. So that denominator is missing a second factor of 2. So we will multiply the numerator and denominator of that fraction by 2. Similarly, the second denominator has a "hole" for the factor of 3. So we will multiply the numerator and denominator of the second fraction by 3:
giving us:
Note that the denominators remain factored. This is on purpose. The denominators are the same so we can add:
Last of all we want to reduce the fraction, if possible. Reducing fraction is done by canceling factors that are common to the numerator and denominator. So we want to see the factors of each. (This is why we left the denominators factored.) Factoring the numerator we get:
And we can see that a factor of 3 will cancel:
leaving
which simplifies to:
or