1. If positive numbers $m, n$ satisfy $m+4 \sqrt{m n}-$ $2 \sqrt{m}-4 \sqrt{n}+4 n=3$, then $\frac{\sqrt{m}+2 \sqrt{n}}{\sqrt{m}+2 \sqrt{n}+3}=$
$$
=1 . \frac{1}{2} \text {. }
$$

Let $\sqrt{m}+2 \sqrt{n}=t$, then the given equation can be transformed into
$$
\begin{array}{l}
t^{2}-2 t-3=0 . \\
\because t>0, \therefore t=3 .
\end{array}
$$

Thus, $\frac{\sqrt{m}+2 \sqrt{n}}{\sqrt{m}+2 \sqrt{n}+3}=\frac{t}{t+3}=\frac{1}{2}$.