The adjacent coefficients have the same ratio suggesting you start by factoring each of those groups.
... = (z^5 -3z^4) -(16z -48)
... = z^4(z -3) -16(z -3)
... = (z^4 -16)(z -3)
The first of these factors is the difference of squares, so can be factored accordingly.
... a² -b² = (a-b)(a+b)
... = (z^2 -4)(z^2 +4)(z -3)
Again, the first of these factors is the difference of squares, so we can continue ...
... = (z -2)(z +2)(z^2 +4)(z -3) . . . . . complete factorization in real numbers
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In the complex numbers, z²+4 is also a difference of squares, you can factor that quadratic to two more linear terms.
.. = (z +2)(z -2)(z -3)(z +2i)(z -2i) . . . . . factorization over complex numbers