The condition for  $3x^2+2pxy+2y^2+2ax-4y+1$  can be resolved into two linear factors is 

$\begin{array}{l}3x^2+2pxy+2y^2+2ax-4y+1\\ a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0\\ a=3,2h=2p,b=2;2g=2a;\\ 2f=-4,c=1,h=p,g=a,f=-2,c=1\end{array}$

$\begin{array}{l}abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0\\ \left(3\right)\left(2\right)\left(1\right)+2(-2)\left(a\right)\left(p\right)-3\left(4\right)-2{\left(a\right)}^2-1\left(p^2\right)=0\\ 6-4pa-12-2a^2-p^2=0\\ -p^2-2a^2-4ap-6=0\\ p^2+2a^2+4ap+6=0\end{array}$