Question

Find the roots of x^3 - x + 2x^2 - 2 = 0

Answer 1

you want to try values of x such as 1,2,3,4,5,6 until it gives 0
x=1

1-1+2-2
hence we get 3-3=0

so 1 is a solution so we have

x=1, x-1=0
so (x-1)(ax^2+bx+c)=x^3-x+2x^2-2

Answer 2

so when we multiply this out we get

ax^3+bx^2+cx-ax^2-bx-c=x^3-x+2x^2-2

so we have

ax^3+bx^2-ax^2+cx-c=x^3-x+2x^2-2

compare coefficients so
for x^3

a=1

for x^2
b-a=2

for constants
-c=-2
c=2

so b-a=2, a=1, b-1=2, b=3

so we have

(x-1)(x^2+3x+2)=x^3-x+2x^2-2