On the cubic equation with five unknowns 3(x³ - y³) = z³ - w³ + 12t² + 4

The non-homogeneous Cubic Equation with five unknowns represented by 3(x³ - y³) = z³ - w³ + 12t² + 4 is analyzed for its patterns of non–zero integral solutions. Six different patterns of non-zero distinct integer solutions are obtained. A few interesting properties between the solutions and special number patterns namely Polygonal numbers, Centered Polygonal numbers, Pyramidal numbers, Stella Octangular numbers, Star numbers and Pentatope number are exhibited.