1. **State the problem:** Simplify or factor the expression $8x^{12} - 27y^{15}$. 2. **Recognize the form:** This is a difference of cubes because $8x^{12} = (2x^4)^3$ and $27y^{15} = (3y^5)^3$. 3. **Formula for difference of cubes:** $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 4. **Apply the formula:** Let $a = 2x^4$ and $b = 3y^5$. $$8x^{12} - 27y^{15} = (2x^4 - 3y^5)((2x^4)^2 + (2x^4)(3y^5) + (3y^5)^2)$$ 5. **Simplify inside the second parentheses:** $$= (2x^4 - 3y^5)(4x^8 + 6x^4y^5 + 9y^{10})$$ 6. **Final factored form:** $$8x^{12} - 27y^{15} = (2x^4 - 3y^5)(4x^8 + 6x^4y^5 + 9y^{10})$$