[tex]\displaystyle\bf\\\left \{ {{a_{6} =14} \atop {a_{10} +a_{15} =80}} \right. \\\\\\\left \{ {{a_{1} +5d=14} \atop {a_{1} +9d+a_{1} +14d=80}} \right.\\\\\\\left \{ {{a_{1} +5d=14} \ |\cdot(-2) \atop {2a_{1} +23d=80}} \right. \\\\\\+\left \{ {{-2a_{1} -10d=-28} \atop {2a_{1} +23d=80}} \right. \\------------\\13d=52\\\\d=4\\\\a_{1} =14-5d=14-5\cdot 4=14-20=-6\\\\\\a_{12} =a_{1} +10d=-6+10\cdot 4=-6+40=34\\\\\boxed{a_{12} =34}[/tex]

[tex]\displaystyle\bf\\S_{10} =\frac{2a_{1} +9d}{2} \cdot 10=(2a_{1}+9d)\cdot 5 =\Big[2\cdot (-6)+9\cdot 4\Big]\cdot 5=\\\\\\=\Big(-12+36\Big)\cdot 5=24\cdot 5=120\\\\\\\boxed{S_{10} =120}[/tex]