\begin{array}{l}
 p \cdot \frac{{q\left( {3 - q} \right)}}{{p^3 }} = \frac{{\left( {1 - p} \right)\left( {2 + p} \right)}}{{p^2 }} = 1 \Rightarrow \\ 
 \Rightarrow 3 - p - p^2 = p^2 \Rightarrow 2p^2 + p - 3 = 0 \Rightarrow \\ 
 \Rightarrow p = \frac{{ - 1 + \sqrt {1 + 24} }}{4} = 1 \\ 
 \end{array}