Answer:
(x -1)³(x +1) = x⁴ -2x³ +2x -1
Explanation:
You want the least common multiple of (1 -x²) and (x -1)³.
LCM
The least common multiple (LCM) will be the product of the unique factors, each to its highest power.
1 -x² = -(x -1)(x +1)
(x -1)³ = (x -1)³
The unique factors to their highest powers are ...
(x -1)³ and (x +1)
The LCM is (x -1)³(x +1) = x⁴ -2x³ +2x -1.
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Additional comment
You may or may not want to add a minus sign in front of the LCM expression. We prefer to leave it with a positive leading coefficient. For whatever sign you choose, the leading coefficient of the quotients when divided by these expressions will have a minus sign on one or the other.