After checking for easy solutions such as completing the square, I resorted to long division. Although it takes several steps, it does work.
We know that x+1, x-1, x+2, and x-2 are the most likely roots, because they are the factors of -2. So we start with one of these.
I confess that I started with x-2 and it does not work. Next I tried x+2 as shown below.
`x^4 + 3x^3+x^2-3x -2` divided by `x+2`
We need the `x^3` because that gives us `x^4` and `2x^3`
We need one more `x^3` that is why we have `x^2`
That gives us `3x^3` and `2x^2`
We only need one `x^2` so We need a `-x`
Now we have `-2x` and we need `-1x` , so we subtract one.
With the `-1` in the second factor, we know to try `x+1` or `x-1` next.
I chose `x+1` because I could see that it would give me a positive` ``x^2`` `
The last term is the difference of squares, so it is
Setting each of the factors equal to 0 we get
The roots are ` x=-2 ` , ` ``x=-1`` ` , ` ``x=-1`` ` , and `x=1`
Roots are -2, -1, and 1