Express (4x)/((x^2) - 9) - (2)/(x + 3) as a single fraction in its simplest form (4 marks)

To do this we need to find a common denominator. We have an (x + 3) on the right hand side and a difference of two squares ((x^2)-9) on the left hand side. So factorising what looks more complicated is first priority. The difference of two squares factorises into (x + 3)(x - 3) (this gets the first mark). Now we have (x + 3) in both denominators, but no (x - 3) on the right side. Therefor to get this we must times the right side by (x - 3)/(x - 3) to get (2(x - 3))/(x + 3)(x - 3). We can then combine the two fractions to make one fraction of (4x - 2(x - 3))/(x - 3)(x + 3) (second mark achieved here)
We now have it as a single fraction! However it has asked for our simplest form. First we should expand the numerator as we have an x inside and outside the bracket - this gives us: (4x - 2x + 6)/(x - 3)(x + 3). This then becomes: (2x + 6)/(x - 3)(x + 3) (third mark!). On the numerator we now have a 2x and a 6, these are both multiples of 2, meaning we can take the 2 out, making it: (2(x + 3))/(x + 3)(x - 3). Finally with an (x + 3) on the top and bottom of the fraction we can cancel them out leaving us with our single fraction in its simplest form: 2/(x - 3) (final answer mark!).