Prove it
Let $a,b,c,A,B,C\in \mathbb{R},a\neq 0,A\neq 0.$ For every real number $x$,$\left | ax^{2}+bx+c \right |\leq \left | Ax^{2}+Bx+C \right |$. Prove that \[\left | b^{2}-4ac \right |\leq \left | B^{2}-4AC \right |\]
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$