(2/z-1) - 1/4 = 3/(z+1)

this is what you have asked for is it what you want?

$$\left({\frac{{\mathtt{2}}}{{\mathtt{z}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right){\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}} = {\frac{{\mathtt{3}}}{\left({\mathtt{z}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}}$$

anyway if it is firstly recognise that z cannot be 0 or -1

Simplify the LHS

then

multiply both sides by 4z(z+1)

that will get rid of all the fractions and make it easy to work with.

give it a go and if you get stuck or need more help then show us what you've done or ask us to help more.