Let’s simplify the equation,

(x² + 1)² - x² = 0

⇒ x⁴ + 2x² + 1 - x² = 0

⇒ x⁴ + x² + 1 = 0

Let x² = y,

⇒ y² + y + 1 = 0

D = b² - 4ac = 0

= 1² - 4(1)(1)

∴ D = -3 < 0

Hence, the equation (x² + 1)² - x² = 0 has no real roots.