The distance between $(0,0)$ and the tangent to the curve $y=e^{2x}+x^2$ drawn at  $x=0$, is -

$$\begin{gathered} y=e^{2x}+x^2 \text{at }x=0, y=1 \\ \frac{dy}{dx}=2e^{2x}+2x \\ \frac{dy}{d{x}_{(0,1)}}=2 \end{gathered}$$

equation of tangent is  $y-1=2(x-0)$
i.e.  $2x-y+1=0$
so distance from origin to also line is  $d=\frac{1}{\sqrt{5}}$