A few comments:

• The symmetry of the problem leads to 10 different cases only. We can assume that 1<=$c<$b<$a<=$rep.
• This also implies that $a is always at least 3, and $b at least 2.
• This way the loops are simpler and no check for equality is required.
• I think that $a, $b, $c are the correct parameters to store the results rather than the polynomials itself.

Here is my proposed code:

use strict;
use warnings;
use Math::Polynomial::Solve qw!poly_derivative poly_roots!;
use Data::Dumper;

my %haystack;
my $rep = int 5; # $rep means right endpoint of the interval [0, $rep]

foreach my $a (3..$rep ) {
    foreach my $b (2..$a-1 ) {
        foreach my $c (1..$b-1 ) {
      my @quintic = (1, -$a - $b - $c, $a*$b + $a*$c + $b*$c, -$a*$b*$
+c, 0, 0); 
      my @derivative = poly_derivative(@quintic);
      my @zeros = poly_roots(@derivative);    
      
      $haystack{"$a,$b,$c"}{"p"} =  \@quintic;
      $haystack{"$a,$b,$c"}{"d"} =  \@derivative;
      $haystack{"$a,$b,$c"}{"z"} =  \@zeros;
        }
    }
}

print Dumper{%haystack};
[download]

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