Without actual division, prove that `x^4+2x^3-2x^2+2x-3` is exactly divisible by `x^2+2x-3.`

Explanation

Let `p(x)=x^4+2x^3−2x^2+2x−3`
Let `g(x)=x^2+2x−3`
Then, `g(x)=x^2+2x−3`
`=x^2+3x−x−3`
`=x(x+3)−(x+3)`
`=(x+3)(x−1)`
Now, we check if `g(x)` is a factor of `p(x)` by using factor theorem.
`∴ (x+3)` and `(x−1)` divides `p(x)` if `p(−3)` and `p(1)=0`
So, `p(−3)=(−3)^4+2(−3)^3−2(−3)^2+2(−3)−3`
`=81−54−18−6−3=0`
and, `p(1)=(1)^4+2(1)^3−2(1)^2+2(1)−3`
`=1+2−2+2−3=0`
Hence, `p(x)` is divisible by `(x+3)` and `(x−1)`
`⇒p(x)` is divisible by `(x+3)(x−1)`
`⇒p(x)` is divisible by `g(x)`
Hence proved.