Show that `x+1` and `2x-3` are factors of `2x^3-9x^2+x+12`

Explanation

Let the given polynomial be `p(x)=x+2x^3-9x^2+12`
Let the given factor be `g(x)=x+1`
`x=−1`
Now,
`p(−1)=2(−1)^3−9(−1)^2+(−1)+12`
`=−2−9−1+12`
`=0`
`therefore x+1` is the factor of `p(x)`
Again,
Let the given polynomial be `p(x)=x+2x^3-9x^2+12`
Let the given factor be `g(x)=2x - 3`
Let `g(x) = 0`
`2x-3=0`
`2x=3`
`x=3/2`
Substitute `x=3/2` in `p(x)`,
`p(3/2)=(3/2)+2(3/2)^3-9(3/2)^2+12`
`=3/2+2(27/8)-9(9/4)+12`
`=3/2+27/4-81/4+12`
`=(3(2)+27-81+12(4))/4`
`=(6+27-81+48)/4`
`=(33+48-81)/4`
`=(81-81)/4`
`=0/4`
`=0`
`therefore 2x-3` is the factor of `x+2x^3-9x^2+12`
`therefore x+1` and `2x-3` are the factors of `x+2x^3-9x^2+12`