Answer: (x + 1)(x - 2)(x - i√2)(x + i√2)
Explanation:
f(x) = x⁴ - x³ - 2x - 4
Possible rational roots are: ±1, ±2, ±4
Use synthetic division to find one of the roots. If the remainder is zero, then it is a root:
-1 | 1 -1 0 -2 -4
| ↓ -1 2 -2 4
1 -2 2 -4 0 ← remainder is zero so (x + 1) is a factor
Use synthetic division on the factored polynomial to find one another one of the roots.
2 | 1 -2 2 -4
| ↓ 2 0 4
1 0 2 0 ← remainder is zero so (x - 2) is a factor
Use sum & difference of a square on the factored polynomial:
x² + 2
= x² - (-2)
= (x - i√2)(x + i√2)