An eighth-degree polynomial in three positive variables

Prove the inequality: $g = \left( y+z \right) ^{2}{x}^{6}+ \left( y+z \right) ^{3}{x}^{5}+yz \left( 3\,{z}^{2}+3\,{y}^{2}-58\,yz \right) {x}^{4}+ \left( {z}^{2}+4\,yz+{y}^{2} \right) \left( y+z \right) ^{3}{x}^{3}+ \left( {z}^{4}+{z}^{3}y+{y}^{3}z+{y}^{4}+3\,{y}^{2}{z}^{2} \right) \left( y+z \right) ^{2}{x}^{2}-yz \left( 2\,{z}^{2}+3\,yz+2\,{y}^{2} \right) \left( y+z \right) ^{3}x+{y}^{2}{z}^{2} \left( {y}^{2}+{z}^{2}+3\,yz \right) \left( y+z \right) ^{2}\geq 0$ for $x, y, z > 0$.

Source: InternLM Lean-Workbook, record lean_workbook_46677 (Apache-2.0). Complete source proposition preserved; proof developed independently.