Let f(x) = x⁴ – 4x³ + 12x² + x – 1

Let a, b, g, d are the root of equation.

 $\therefore \mathrm{\alpha \beta \gamma \delta }$ = – 1 so the equation has at least two real roots.                             ………..(ii)

f’(x) = 4x³ – 12x² + 24x + 1

f’’(x) = 12x² – 24x + 24 = 12((x + 1)² + 1)

so f’’(x) > 0 so f’(x) = 0 has only one real roots so f(x) = 0 has at most two real roots.        ……(ii)

from (i) & (ii)

f(x) = 0 has exactly two real roots.