$t=3^x,A=3^a,B=3^b$
$3^{3x+1}-3^{2x+2}-3^{x+1}+1=3(t-A)(t-B)(t-C)$
$\begin{array}{ccc}
A+B+C&=&3\\
AB+BC+CA&=&-1\\
ABC&=&-\dfrac{1}{3}
\end{array}$
$\begin{array}{rcl}
P&=&\dfrac{3^a-3^b}{1+3^{a+b+1}}\\
&=&\dfrac{A-B}{1+3AB}\\
\\
P^2&=&\dfrac{(A+B)(A+B)-4AB}{1+6AB+9A^2B^2}\\
&=&\dfrac{(A+B)(3-C)-4AB}{(1+AB)(1+9AB)-4AB}\\
&=&\dfrac{(A+B)(-9ABC-C)-4AB}{(-BC-CA)(1+9AB)-4AB}\\
&=&1
\end{array}$