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

$f\left(x\right)={\sin }^{-1}x+2{\tan }^{-1}x+{\left(x+2\right)}^2-3$

$\text{Domine of f}\left(x\right) is \left[-1,1\right]$

$\text{also f}\left(x\right) \text{is incresing function in the domain}$   $\left(\sin ce f\text{'}\left(x\right) >0 for all x\in \left[-1,1\right]\right)$

$p=f_{min}\left(x\right)$

$=f\left(-1\right)=\frac{-\pi }{2}+2\left(\frac{-\pi }{4}\right)+1-3=-\pi -2$

$and q=f_{max}\left(x\right)$

$=f\left(1\right)=\frac{\pi }{2}+2\left(\frac{\pi }{4}\right)+9-3=\pi +6$

$Range of f\left(x\right) is \left[-\pi -2,\pi +6\right]$ =$\left[p,q\right]$

$hence \left(p+q\right) =4$