Domain, Max and Min

$\\ \\ f\left( x \right) ={ x }^{ 2 }+2x+a\\ f^{ ' }\left( x \right) =2x+2\\ f^{ ' }\left( x \right) \quad is\quad equal\quad to\quad zero\quad for\quad min\quad or\quad max\quad value\quad of\quad f\left( x \right) \\ but\quad as\quad coefficient\quad of\quad { x }^{ 2 }\quad is\quad greater\quad than\quad zero,\quad f^{ ' }\left( x \right) \quad is\quad zero\quad for\quad min\quad value\quad of\quad f\left( x \right) \\ \therefore \quad 2x+2=0\quad \Rightarrow \quad x=-1\\ f\left( -1 \right) =\quad { f\left( x \right) }_{ min }\quad =-1\\ ie,\quad { (-1 })^{ 2 }\quad +2(-1)\quad +a\quad =-1\\ \Rightarrow \quad a=0\\ \therefore \quad f\left( x \right) ={ x }^{ 2 }+2x\\ Now,\quad { f\left( x \right) }_{ max }=63\\ ie,\quad { x }^{ 2 }+2x=63\\ \Rightarrow x=-9,7\\ as\quad x\le 0,\quad x=-9\\ \therefore \quad b=-9\\ so,\quad a-b=\quad 0-(-9)\quad =\boxed { 9 }$