Finding polynomial coeffecients

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If (x+2) is a factor of x^3+px^2-x-2 find p and the other factors when p has this value.

I used synthetic division and identified that 4p+2x-10=0; 4p+2x=10

HENCE

2p+x=5

Beyond that however, I am lost as to how I would go about finding the values of either P or X. My confusion stems from the fact that I have only been given one factor, namely, (x+2) and so it would appear that I am not able to utilise simultanteous equations to find the value of one term then substitute it into the equation to find the value of the other.

I don't think you're supposed to use simultaneous equations
I used synthetic division and directly found out that P=2
And using that value and I found the other factor to be x^2-1

How did you find that p=2?

When I used synthetic division, I used coeffecients 1 P -X and -2 with -2 as the input.

so, 1, p-2; -2+4-x 4p-10+2x

p=2; (x+1)(x-1)

Is the answer given

So yes, your correct; which I had suspected from the beginning. When I gave my response, I was showing my working to identify where I had made a mistake

Alright. The thing is I'm not sure how you arrived to those figures. You should directly get p=2 if you put in -2 as the input and your other inputs are correct.
You're final -8+4p=0
0 because they've stated it's a factor so you can just insert that and directly find P
Try and see if that helps

Did you include X as a coeffecient?

Oh, I just figured out your mistake!
You mentioned one of your values as -x
that is incorrect as you are supposed to mention only the coefficients not x. So the coefficient is -1

Thank you! . Can't believe I did that :|