Proof that a quadratic graphs as parallel lines.

Question

I define parallel lines in the Euclidean plane as two lines with a constant separation between points closest to each other on opposite lines. We might also, just say that lines are parallel when they share proportional direction vectors. To prove that two lines are parallel, we can try to solve them simultaneously and arrive at a contradiction. If we take two (or more) line equations and multiply them together, we get a quadratic equation (or higher power) which when graphed continues to be two parallel lines. For example: $L_1:\;y=2x-3\qquad L_2:\;y=2x+1$ These two equations can be expressed as $$4 \; x^{2} - 4 \; x \; y - 4 \; x + y^{2} + 2 \; y = 3$$ These will factor into $L_1$ and $L_2$ $$(y-2x+3)(y-2x-1)=0.$$ What theorem or axioms allows me to separate these factors into linear equations and continue with a contradiction proof that the quadratic represents parallel lines? Is it even true that we can multiply linear equations together in great hordes and obtain large numbers of parallel lines from a high power polynomial?