code: def dmp_mul(f, g, u, K):
      """
      Multiply dense polynomials in ``K[X]``.
  
      Examples
      ========
  
      >>> from sympy.polys import ring, ZZ
      >>> R, x,y = ring("x,y", ZZ)
  
      >>> R.dmp_mul(x*y + 1, x)
      x**2*y + x
  
      """
      if not u:
          return dup_mul(f, g, K)
  
      if f == g:
          return dmp_sqr(f, u, K)
  
      df = dmp_degree(f, u)
  
      if df < 0:
          return f
  
      dg = dmp_degree(g, u)
  
      if dg < 0:
          return g
  
      h, v = [], u - 1
  
      for i in range(0, df + dg + 1):
          coeff = dmp_zero(v)
  
          for j in range(max(0, i - dg), min(df, i) + 1):
              coeff = dmp_add(
                  coeff, dmp_mul(f[j], g[i - j], v, K), v, K
              )
  
          h.append(coeff)
  
      return dmp_strip(h, u)
  
imports: from sympy.polys.densebasic import dup_slice, dup_LC, dmp_LC, dup_degree, dmp_degree, dup_strip, dmp_strip, dmp_zero_p, dmp_zero, dmp_one_p, dmp_one, dmp_ground, dmp_zeros
  from sympy.polys.polyerrors import ExactQuotientFailed, PolynomialDivisionFailed
label_desc: Use image node_id 32 for calling a global function with example usage: dmp_mul(f, g, u, K) and returns: dmp_strip, dup_mul, dmp_sqr, f, g