How to solve this system of equations systematically? This might seem a trivial problem, but I have some trouble in arranging the data. So suppose you are given $f(x,y)=x^2y^2(1+x+2y)$ and you want to find it's critical points. Thus we find
$$\frac{\partial f}{\partial x}(x,y)=xy^2(2+3x+4y)\textrm{ and }\frac{\partial f}{\partial y}(x,y)=2x^2y(1+x+3y).$$
Now we set $f_x= 0$ and $f_y=0.$  Thus we get a system 
$$
\begin{split}
xy^2(2+3x+4y) &=0\\
2x^2y(1+x+3y) &=0
\end{split}
$$
Now we have a bunch of cases. The way I think about this is as follows:
$$((x=0)\lor(y=0)\lor(3x+4y=-2))\land((x=0)\lor(y=0)\lor(x+3y=-1)).$$
Then I consider each possibility separately, but this seems to be slow and sometimes I forget some solutions. Thus I was wondering if there are any other methods which one can use to solve such type of problems. 

Use the converse of the distributive property:
$((x=0)\lor(y=0)\lor(3x+4y=-2))\land((x=0)\lor(y=0)\lor(x+3y=-1))\\\equiv(x=0)\lor(y=0)\lor[(3x+4y=-2)\land(x+3y=-1)]$
$3x+4y+2=0=x+3y+1$ is just a pair of straight lines (linear equations) intersecting at $(-2/5,-1/5)$. Therefore, you have $(x=0)\lor(y=0)\lor(x=-2/5\land y=-1/5)$
