(z^2+36)/(z^2-36)=5/(z+6)+z(z-6) The division symbol represent fractions
You should first of all factorise the left hand side of the equation so that it is terms of (z + 6) and (z - 6). Once you have done that you should make sure that each of your 3 terms have a denominator of (z + 6)(z - 6) so you can put them all together and solve.
Have a go at that and see how far you get.