If $\alpha ,\beta$ are the distinct roots of $x^2+bx+c=0$, then  $\underset{x\to \beta }{\lim }\frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{{(x-\beta )}^2}$ is equal to

$L=\underset{x\to \beta }{\lim }\frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{{(x-\beta )}^2}$

$L=\underset{x\to \beta }{\lim }\frac{[e^{2(x^2+bx+c)}-1-2(x^2+bx+c)]{(x-\alpha )}^2}{{(x-\alpha )}^2{(x-\beta )}^2}$

$\begin{array}{l}=2{(\beta -\alpha )}^2\\ =2[{(\beta +\alpha )}^2-4\alpha \beta ]=2(b^2-4c)\end{array}$