Number of terms

  1 term x 1 term -- (2x)(3x)
  1 term x 2 terms -- (2x)(3x + 4)
  1 term x 3 terms -- (2x)(3x² + 4x + 5)
  2 terms x 2 terms -- (2x + 3)(4x + 5)
  2 terms x 3 terms -- (2x + 3)(4x² + 5x + 6)
  3 terms x 3 terms -- (2x² + 3x + 4)(5x² + 6x + 7)

Coefficient can be Zero?

  (2x2 + 3x + 4) NO
  (2x4 + 3x2 + 4) YES

Method of working the problem

Horizantal
Horizontal, plus show the distributions
Vertical