Question

let P(x)=2x^4-x^3+13^2-8x-24. Use synethetic division to show that x+1 is a factor of P(x)

Answer:2x^3-3x^2+16x-24

Then find all real and imaginary roots of P(x)=0.

Answer 1

Have you learned about synthetic (or polynomial) division?

If not, you should. The following reference should help.

One of the roots of P(x) = 0 will be x = -1, since x+1 is a factor. To get the other three roots, you need to solve the cubic equation
2x^3 -3x^2 +16x -24 = 0

That will not be easy.
You will find some hints here:
http://oakroadsystems.com/math/polysol.htm