$$\begin{aligned} & 2 x-y+3=0 \\ & 6 x+3 y+1=0 \\ & \alpha x+2 y-2=0 \end{aligned}$$

Will not form a $$\Delta$$ if $$\alpha x+2 y-2=0$$ is concurrent with $$2 x-y+3=0$$ and $$6 x+3 y+1=0$$ or parallel to either of them so

Case-1: Concurrent lines

$$\left|\begin{array}{ccc} 2 & -1 & 3 \\ 6 & 3 & 1 \\ \alpha & 2 & -2 \end{array}\right|=0 \Rightarrow \alpha=\frac{4}{5}$$

Case-2 : Parallel lines

$$\begin{aligned} & -\frac{\alpha}{2}=\frac{-6}{3} \text { or }-\frac{\alpha}{2}=2 \\ & \Rightarrow \alpha=4 \text { or } \alpha=-4 \\ & P=16+16+\frac{16}{25} \\ & {[P]=\left[32+\frac{16}{25}\right]=32} \end{aligned}$$