Find `alpha\ a n d\ beta` , if `x+1\ a n d\ x+2` are factors of `x^3+3x^2-2alphax+beta`

Explanation

`p(x)=x^3+3x^2−2αx+β(x+1)` and `(x+2)` are the factors of `p(x)`
`=>p(−1)` and `p(−2)=0`
`∴p(1)=(−1)^3+3(−1)^2\2(1)+=0`
`⇒−1+3+2α+β=0`
`⇒β=−2a−2` ......(i)
`p(−2)=(−2)^3+3(−2)^2−2α(−2)+β=0`
`⇒−8+12+4α+β=0`
`⇒β=−4α−4` .....(ii)
From equation (i) and equation (ii)
`=>−2α−2=−4α−4`
`⇒2α=−2`
`⇒α=−1`
Put `α=−1` in equation (1)
`⇒β=−2(−1)−2=2−2=0`
`therefore α=−1`and `β=0`