The number of distinct real roots of x⁴ - x³ + 12x² + x - 1 = 0 is

f(x) = x⁴ - x³ + 12x² + x - 1
Let f(x) has four distinct real roots ⇒ f'(x) = 4x³ - 3x² + 24x + 1
f'(x) has three distinct real roots
f''(x) = 12x² - 6x + 24 = 12(x² - 1/2x + 2)
D = 1/4 - 4(2) < 0
f''(x) cannot have 2 real solutions. So, f(x) cannot have four real distinct roots
It can have 2 or 0 real roots.
f(0) = -1, f(1) = 9 ⇒ At least one real solution
So, 2 real distinct solutions.