Найдите произведение корней уравнения (х – 1)⁴ - x² + 2x - 73 = 0.

Let y = x - 1. Then x = y + 1. The equation becomes y⁴ - (y+1)² + 2(y+1) - 73 = 0. Expanding this, we get y⁴ - (y² + 2y + 1) + 2y + 2 - 73 = 0, which simplifies to y⁴ - y² - 72 = 0. Let z = y². Then z² - z - 72 = 0. Factoring this quadratic, we get (z - 9)(z + 8) = 0. So, z = 9 or z = -8. This means y² = 9 or y² = -8. Thus, y = ±3 or y = ±2i√2. Substituting back x = y + 1, we get x = 1 ± 3 (so x = 4, -2) and x = 1 ± 2i√2. The roots are 4, -2, 1 + 2i√2, 1 - 2i√2. The product of the roots is 4 * (-2) * (1 + 2i√2) * (1 - 2i√2) = -8 * (1 - (2i√2)²) = -8 * (1 - (-8)) = -8 * 9 = -72.