Question

How do you factor completely `P(x)=x^3+2x^2-x-2`?

Answer 1

`(x-1)(x+2)(x+1)`

Explanation:

Spotting that `1+2-1-2=0` means that `x-1` is a factor.
Then write `x^3+2x^2-x-2-=(x-1)(...x^2 ... x ...)` and try to fill in the gaps to make the identity true by matching powers. Obviously the coefficient of `x^2` has to be `1`, and similarly the constant at the end has to be `+2`:

`(x-1)(x^2....x+2)`

Since we have got `+2x` and need `-x` we need a `+3x` added in:
`(x-1)(x^2+3x+2)` which checks out upon multiplying out.
The second bracket then factorizes easily to `(x+2)(x+1)`.