Stuck on polynomial question

stuck on this question
The polynomial P(x) has factors x - 1 and x + 2 where

P(x)= x^3 + cx^2 + dx - 8

and c, d are constants

what is the value of c and what is the value of d
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also express P(x) as the product of linear factors.

could you please show your working and explain steps thanks

I got that c is 5 and d is 2 by substituting in x=1 and x=-2 and using simultaneous equations

then I used x^3 + 5x^2 + 2x - 8 = (x-1)(Ax^2 + Bx + C)

A = 1
To work out B, you use the fact that the coefficient of x^2 of the polynomial is 5
5= -A+B
B=6
C=8

sub them all in
(x^2 + 6x + 8) = 0
you know that one factor is (x+2)
The other is (x+4)
Because (x+2)(x+4) = (x^2 + 6x + 8)

so expressing it as linear factors:
(x+2)(x+4)(x-1)

P(1) = 0 and P(-2) = 0 gives you two equations in c,d. Solve them. Then find third root to express P(x) as a product of three linear factors.