If $m+n=5$ and $mn=3$, find $\sqrt{\frac{n+1}{m+1}} + \sqrt{\frac{m+1}{n+1}}$? It is known that $m+n=5$ and $mn=3$. So what is the value of:

$$
\sqrt{\dfrac{n+1}{m+1}} + \sqrt{\dfrac{m+1}{n+1}}
$$

I think we're suppose to solve for the system of equations first, but I'm not getting any results that's useful.

\begin{align}
\sqrt{\frac{n+1}{m+1}}+\sqrt{\frac{m+1}{n+1}}&=\frac{(n+1)+(m+1)}{\sqrt{(m+1)(n+1)}}
\\ \\
&=\frac{m+n+2}{\sqrt{mn+m+n+1}}
\\ \\
&=\frac{5+2}{\sqrt{3+5+1}}
\\ \\
&=\frac{7}{3}
\end{align}
