If range of the function $\mathrm{f}\left(\mathrm{x}\right)={\sin }^{-1}\mathrm{x}+2{\tan }^{-1}\mathrm{x}+{\mathrm{x}}^2+4\mathrm{x}+1\text{ is }[\mathrm{p},\mathrm{q}]$ then the value of (p + q) is

$\mathrm{f}\left(\mathrm{x}\right)={\sin }^{-1}\mathrm{x}+2{\tan }^{-1}\mathrm{x}+(\mathrm{x}+2{)}^2-3$
$\text{ Domain of }\mathrm{f}\left(\mathrm{x}\right)\text{ is }[-1,1]$ Also f(x) is an increasing function in the domain
$\begin{array}{l}\therefore \mathrm{p}={\mathrm{f}}_{min}\left(\mathrm{x}\right)=\mathrm{f}(-1)\\ =-\frac{\mathrm{\pi }}{2}+2\left(\frac{-\mathrm{\pi }}{4}\right)+1-3=-\mathrm{\pi }-2\text{ and }\\ \mathrm{q}={\mathrm{f}}_{max}\left(\mathrm{x}\right)=\mathrm{f}\left(1\right)=\frac{\mathrm{\pi }}{2}+2\left(\frac{\mathrm{\pi }}{4}\right)+9-3=\mathrm{\pi }+6\end{array}$
Therefore, the range of f(x) is $[-\mathrm{\pi }-2,\mathrm{\pi }+6]$