We know that sum of probabilities of all exhaustive events for a sample space is $1$.

$\Rightarrow p^2+2p+4p-1=1$

$\Rightarrow p^2+6p-2=0$

$\Rightarrow p=\frac{-6\pm \sqrt{6^2-4\left(1\right)\left(-2\right)}}{2\left(1\right)}=-3\pm \sqrt{11} ...\left(\mathrm{i}\right)$

We also know that probability of an event belongs to $\left[0,1\right]$.

$\Rightarrow 0\le p^2\le 1 \& 0\le 2p\le 1 \& 0\le 4p-1\le 1$

$\Rightarrow -1\le p\le 1 \& 0\le p\le \frac{1}{2} \& \frac{1}{4}\le p\le \frac{1}{2}$

$\Rightarrow \frac{1}{4}\le p\le \frac{1}{2} ...\left(\mathrm{ii}\right)$

From equations $\left(\mathrm{i}\right) \& \left(\mathrm{ii}\right)$,

$p=\sqrt{11}-3$