[tex]\displaystyle\bf\\A(1 \ ; \ -4) \ \ \ \Rightarrow \ \ \ x=1 \ \ ; \ \ y=-4\\\\B(-2 \ ; \ 5) \ \ \ \Rightarrow \ \ \ x=-2 \ \ ; \ \ y=5\\\\\\y=x^{2} +bx+c\\\\\\\left \{ {{-4=1^{2} +b\cdot 1+c} \atop {5=(-2)^{2} +b\cdot(-2)+c}} \right. \\\\\\\left \{ {{1+b+c=-4} \atop {4-2b+c=5}} \right. \\\\\\-\left \{ {{b+c=-5} \atop {-2b+c=1}} \right. \\---------\\3b=-6\\\\\boxed{b=-2}\\\\c=-5-b=-5-(-2)=-5+2=-3\\\\\boxed{c=-3}[/tex]