Hi Rory,

First factor the denominators, x²-64 and x²-6x-16. This will show you how to write the two fractions with a common denominator. Do so, add the fractions and simplify.

Writ back if you have any difficulty with the steps.

Harley

As your title suggests, although what you are adding are rational functions not fractions, you use the same steps as in adding fractions.

Necessary steps:

(1) Express both these rational functions using a common denominator.
(2) Add the numerators and put the sum over that common denominator.

Asa common denominator, you may always use the product of the original denominators. However, it it probably better to prepare by factoring them and then using the LCM: that is, the simplest polynomial with all the factors of both denominators.

At the end, it is usual (and useful, and mandatory in some classrooms) to simplify and factor the numerator, and cancel any factors in common with the denominator. (These may well not have been present in the original rational functions.)
This is usually easier if the denominator was already factored, though it may be easier to test a high-degree denominator for divisibility by a specified factor than to "factor it cold".

Good Hunting!
RD