At this sort of level, it's pretty much a case of trial and error.
Substitute values for z and hope that one of them gets you a zero value for the polynomial.
Start off by trying z = 0,1,2,-1,-2,3 etc. For this particular example you get a hit with z = 1 in which case z - 1 is a factor and you can now divide out to determine the quadratic.
Notice though that if for two of your trial values the polynomial changes sign then there has to be a zero between those two values. So for your example z = 0 and z = 2 gets you polynomial values of -2 and +2 in which case there has to be a value of z between 0 and 2 making the poynomial zero. (In general this need not be an integer value.)
If you get no joy from this method you might consider sketching a graph or graphs (usually graphs is better).
For this example rewrite the equation as and sketch the graphs of and The values of z at which the graphs intersect will be the roots of the equation.
For this example this will tell you that there is a root somewhere between 1/2 and 3.