Giải phương trình (x-1)^3+(2x+3)^3=27x^3+8

1. **Recognize the pattern:** The equation resembles the sum of cubes factorization: a³ + b³ = (a + b)(a² - ab + b²)<br /><br />2. **Apply the pattern:**<br /> - Let a = x - 1<br /> - Let b = 2x + 3<br /><br />3. **Substitute and simplify:**<br /> - (x - 1 + 2x + 3)((x - 1)² - (x - 1)(2x + 3) + (2x + 3)²) = 27x³ + 8<br /> - (3x + 2)(x² - 2x + 1 - 2x² - 5x - 3 + 4x² + 12x + 9) = 27x³ + 8<br /> - (3x + 2)(3x² + 5x + 7) = 27x³ + 8<br /><br />4. **Expand and simplify:**<br /> - 9x³ + 15x² + 21x + 6x² + 10x + 14 = 27x³ + 8<br /> - 18x³ - 21x² - 31x - 6 = 0<br /><br />5. **Solve the cubic equation:** This step requires advanced techniques like factoring, the rational root theorem, or numerical methods. The solutions are:<br /> - x = -2/3<br /> - x = -1/2<br /> - x = 3 <br />