To factor it out, we divide each term of the expression by $2x^{2}y^{2}$:
$$\frac{4 x^{2} y^{2} z}{2 x^{2} y^{2}} - \frac{2 x^{5} y^{2}}{2 x^{2} y^{2}} + \frac{6 x^{3} y^{2} z^{2}}{2 x^{2} y^{2}}$$
Simplifying each term, we get:
$$2z - x^{3} + 3xz^{2}$$
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