How do you factor a system of polynomials into their roots the way one can factor a single dimensional polynomial into its roots.

Example

$$x^2 + y^2 = 14$$

$$xy = 1$$

We note that we can find the 4 solutions via quadratic formula and substitution such that the solutions can be separated into $2$ groups of $2$ such that each group lies on either $(x + y + 4)$ and $(x + y - 4)$. Note that:

$$(x + y + 4)(x + y - 4) = 0$$

$$xy = 1$$

Is also an equivalent system.

How do I factor the bottom half?

Ideally if $g$ is a linear expression then my system should be

$$g\_1 \* g\_2 = 0$$

$$g\_3 \* g\_4 = 0$$

Such that the solutions to

Any of the subsystems of this system is a solution to system itself (note there are $4$ viable subsystems).

Help?