Nonnegativity of an expanded sextic polynomial

Prove that $3 a^6+a^5 (15 u+12 v)+a^4 \left(33 u^2+48 u v+15 v^2\right)+a^3 \left(39 u^3+81 u^2 v+48 u v^2+6 v^3\right)+a^2 \left(25 u^4+68 u^3 v+60 u^2 v^2+17 u v^3+v^4\right)+a \left(8 u^5+27 u^4 v+32 u^3 v^2+15 u^2 v^3+2 u v^4\right)+u^6+4 u^5 v+6 u^4 v^2+4 u^3 v^3+u^2 v^4\geq 0$.

Source: InternLM Lean-Workbook, record lean_workbook_15179 (Apache-2.0). Complete source proposition preserved; proof developed independently.