Answer: (x - 5)(x + 5)(x - 2)²
Explanation:
f(x) = x⁴ - 4x³ - 21x² + 100x - 100
Since x = 5, then use synthetic division to find the factored polynomial so we can find another root:
5 | 1 -4 -21 100 -100
| ↓ 5 5 -80 100
1 1 -16 20 0 ← remainder is zero so (x - 5) is a factor
Use the factored polynomial to find another root:
-5 | 1 1 -16 20
| ↓ -5 20 -20
1 -4 4 0 ← remainder is zero so (x + 5) is a factor
Now use the perfect square formula on the factored polynomial:
x² - 4x + 4
= (x - 2)(x - 2)