(x^3-2x^2-4x-4)/(x^2+x-2)
By long division,
Hence,
(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+color(green)((x-10)/(x^2+x-2)
Then, let a and b be unknowns,
color(green)((x-10)/(x^2+x-2))=(x-10)/((x+2)(x-1))
color(white)(xxxxxx//x)=a/(x+2)+b/(x-1)
Multiply throughout by x^2+x-2,
x-10=a(x-1)+b(x+2)
When color(red)(x=1,
color(red)(1)-10=a(color(red)(1)-1)+b(color(red)(1)+2)
color(white)(xxx)3b=-9
color(white)(xxx3)b=-3
When color(blue)(x=-2,
color(blue)(-2)-10=a(color(blue)(-2)-1)+b(color(blue)(-2)+2)
color(white)(....)-3a=-12
color(white)(....-3)a=4
Hence, substitute a=4 and b=-3,
(x^3-2x^2-4x-4)/(x^2+x-2)=x-3+4/(x+2)+3/(x-1)