1. **State the problem:** Factorize the expression $$x^2 - 36y^2 + 2x^3 y^2 - 12 x^2 y^3$$. 2. **Rewrite the expression:** Group terms for easier factorization: $$x^2 - 36 y^2 + 2 x^3 y^2 - 12 x^2 y^3 = (x^2 - 36 y^2) + (2 x^3 y^2 - 12 x^2 y^3)$$. 3. **Factor each group individually:** - First group: $$x^2 - 36 y^2$$ is a difference of squares: $$x^2 - (6y)^2 = (x - 6y)(x + 6y)$$. - Second group: $$2 x^3 y^2 - 12 x^2 y^3 = 2 x^2 y^2 (x - 6 y)$$ by factoring out $$2 x^2 y^2$$. 4. **Combine the factored groups:** The expression becomes: $$(x - 6 y)(x + 6 y) + 2 x^2 y^2 (x - 6 y)$$. 5. **Factor out the common binomial factor $$(x - 6 y)$$:** $$ (x - 6 y) igl( (x + 6 y) + 2 x^2 y^2 igr) = (x - 6 y)(x + 6 y + 2 x^2 y^2)$$. **Final answer:** $$\boxed{(x - 6 y)(x + 6 y + 2 x^2 y^2)}$$