If z is any complex number and iz³ + z² − z + i = 0, where i = √−1, then what is the value of (|z|+1)²?

Factor: iz³ + z² − z + i = z²(iz + 1) − (z − i) = z²(iz+1) + i(iz+1)·... Let me factor differently. iz³ + z² − z + i = iz³ − z + z² + i = z(iz² − 1) + i(z² ... ). Try grouping: iz³ + i + z² − z = i(z³+1) + z(z−1). Hmm. Note iz³ − z = z(iz² − 1), and z² + i. Better: multiply by −i: z³ − iz² + iz + 1 = 0. Factor: z³ + 1 − i(z² − z) = (z+1)(z²−z+1) − iz(z−1) = 0. Trying z = i: i(−i) + (−1) − i + i = 1 − 1 = 0 ✓. So |z| = 1, giving (|z|+1)² = 4.