code: def test_dup_euclidean_prs():
      R, x = ring("x", QQ)
  
      f = (
          x**8
          + x**6
          - 3 * x**4
          - 3 * x**3
          + 8 * x**2
          + 2 * x
          - 5
      )
      g = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
  
      assert R.dup_euclidean_prs(f, g) == [
          f,
          g,
          -QQ(5, 9) * x**4 + QQ(1, 9) * x**2 - QQ(1, 3),
          -QQ(117, 25) * x**2 - 9 * x + QQ(441, 25),
          QQ(233150, 19773) * x - QQ(102500, 6591),
          -QQ(1288744821, 543589225),
      ]
  
file_path: sympy/sympy/polys/tests/test_euclidtools.py
imports: from sympy.polys.rings import ring
  from sympy.polys.domains import ZZ, QQ, RR
  from sympy.polys.specialpolys import f_polys, dmp_fateman_poly_F_1, dmp_fateman_poly_F_2, dmp_fateman_poly_F_3
label_desc: Use image node_id 3 for calling a global function with example usage: test_dup_euclidean_prs() without return types