Some of my students were recently looking at examples like this and wanted some further problems:
Note the thinking behind this:
You should spot that the denominator is the difference of two squares, hence (x+5)(x-5)
If we are to have any hope of simplifying when we factorise the numerator, one of the factors must be (x+5) or (x-5), of those two it must be (x-5) or we would not be able to obtain the -19x term.
Factorise the numerator: (3x-4)(x-5)
A site perhaps not so well known by students is The Centre for Innovation in Mathematics Teaching. If you choose the GCSE course material and scroll down the page you will see all the pupil textbook chapters. (Also note all the other materials for students from 5 – 18!)
Chapter 10 on equations includes many useful algebra examples and exercises including problems of the type above (see worked example 1 on page 41 and the exercises on page 43). Note that each GCSE section has answers.
You should check any factorisation by multiplying out your answer. You could also check answers using WolframAlpha.
Note that you could enter the whole expression and simplify it – WolframAlpha will give you alternate forms; for this type of question you are require to give the simplest single fraction that you can.