1. **Problem statement:** Factor the expression $$14 x^2 y^2 z^2 - 7 xyz + 21 x^3 y^3 z^3$$. 2. **Identify common factors:** Look for the greatest common factor (GCF) in all terms. 3. **Find GCF:** Each term contains a factor of $$7 xyz$$. 4. **Factor out the GCF:** $$14 x^2 y^2 z^2 - 7 xyz + 21 x^3 y^3 z^3 = 7 xyz \left(\frac{14 x^2 y^2 z^2}{7 xyz} - \frac{7 xyz}{7 xyz} + \frac{21 x^3 y^3 z^3}{7 xyz}\right)$$ 5. **Simplify inside the parentheses:** $$= 7 xyz \left(2 xyz - 1 + 3 x^2 y^2 z^2\right)$$ 6. **Final factored form:** $$\boxed{7 xyz (2 xyz - 1 + 3 x^2 y^2 z^2)}$$ This is the fully factored expression by extracting the greatest common factor.