This is a polynomial of the 3rd order, thus x should have three answers. Use the factors given to get values of x as 1, -1 and -2.

Form three equations, and carry out elimination and subsequent substitution to get a = -1/2, b = 1, and c = 1/2

Equate each factor to zero to get the values of x

x-1 = 0, x = 1

x +1 = 0, x = -1

x - 2 = 0, x = 2

substitute these value of x in the given equation

\(ax^3 + bx^2\) + cx - 1

when x = 1

\(ax^3 + bx^2\) + cx - 1 = a + b + c - 1= 0 ------------------------(1)

when x = -1

\(ax^3 + bx^2\) + cx - 1 = -a + b - c - 1 = 0-------------------------(2)

when x = 2

\(ax^3 + bx^2\) + cx - 1 = 8a + 4b + 2c - 1 = 0 ----------------------(3)

add equations 1 and 2

 a + b + c - 1= 0 

+  -a + b - c - 1 = 0

b = 1

put b = 1 into equation (1)

a + b + c - 1= 0 = a + 1 + c -1 = 0

a = - c 

put a = - c into equation(3)

8a + 4b + 2c - 1 = 0, 

-8c + 4 + 2c - 1 = 0 ( b = 1 remember)

-6c = -3

c = \(\frac{-3}{-6}\) = \(\frac{1}{2}\)

put c = \(\frac{1}{2}\) and b = 1 into equation (1)

a + b + c - 1= 0

a + 1 + \(\frac{1}{2}\) - 1 = 0

a + \(\frac{1}{2}\) = 0

then a =  \(\frac{-1}{2}\)