1. **State the problem:** Factor the expression $$147(2x-5)^2 - 48(2y+1)^2$$. 2. **Recognize the formula:** This is a difference of squares, which follows the rule: $$a^2 - b^2 = (a - b)(a + b)$$ where $$a = \sqrt{147}(2x-5)$$ and $$b = \sqrt{48}(2y+1)$$. 3. **Simplify the square roots:** $$\sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}$$ $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$$ 4. **Rewrite the expression:** $$147(2x-5)^2 - 48(2y+1)^2 = (7\sqrt{3}(2x-5))^2 - (4\sqrt{3}(2y+1))^2$$ 5. **Apply the difference of squares formula:** $$= \left(7\sqrt{3}(2x-5) - 4\sqrt{3}(2y+1)\right) \left(7\sqrt{3}(2x-5) + 4\sqrt{3}(2y+1)\right)$$ 6. **Factor out common $$\sqrt{3}$$:** $$= \sqrt{3} \left(7(2x-5) - 4(2y+1)\right) \times \sqrt{3} \left(7(2x-5) + 4(2y+1)\right)$$ 7. **Simplify each factor:** $$= 3 \left(14x - 35 - 8y - 4\right) \left(14x - 35 + 8y + 4\right)$$ 8. **Combine like terms:** $$= 3 (14x - 8y - 39)(14x + 8y - 31)$$ **Final factored form:** $$147(2x-5)^2 - 48(2y+1)^2 = 3 (14x - 8y - 39)(14x + 8y - 31)$$