If $b^2>4ac,$ then $a{(x^2+4x+4)}^2+b(x^2+4x+4)+c=0$ has distinct real roots if ……

$\begin{array}{l}If{x}^2+4x+4=y\\ a{y}^2+by+c=0\because b^2>4acithasrealrootsandaredistinctsayy=\alpha andy=\beta \\ \Rightarrow {(x+2)}^2=y\Rightarrow bothrootsare+ve\end{array}$