Rational zeroes are factors of 32 and can be either positive or negative.
Factors of 32 are 1, 2, 4, 8, 16, 32.
To test if x-1 or x+1 is a factor of the polynomial write the coefficients and place ± in front of the coefficients which have odd powers of x:
x∓6+4±24-32 so we have either 1-6+4+24-32=-9 or 1+6+4-24-32=-45, so neither 1 nor -1 are zeroes of the polynomial.
Now try 2 and -2 for x:
16∓48+16±48-32=0 (for 2 and -2).
2 | 1 -6 4 24 -32
1 2 -8 -8 | 32
1 -4 -4 16 | 0 = x3-4x2-4x+16
-2 | 1 -6 4 24 -32
1 -2 16 -40 | 32
1 -8 20 -16 | 0 = x3-8x2+20x-16.
So we have two lower degree polynomials with rational zeroes: 2, 4, 8, 16.
So we can try ±2 again:
x3∓4x=8-8=0 or -8+8=0 in each case, and -4x2+16=0 so we reduce the polynomial further:
2 | 1 -4 -4 16
1 2 -4 | -16
1 -2 -8 | 0 = x2-2x-8 = (x-4)(x+2).
Now we have all 4 zeroes: x=4, x=-2, x=2 (twice).
Polynomial is (x-4)(x+2)(x-2)2.