Two roots of the polynomial $x^4+x^3-7x^2-x+6$ are $2$ and $-3$. Find two other roots. I have divided this polynomial first with $(x-2)$ and then divided with $(x+3)$ the quotient. The other quotient I have set equal to $0$ and have found the other two roots. Can you explain to me if these actions are correct and why?

Your factorisation 
$(x-2)(x+3)(x-1)(x+1)$ 
is zero if $x \in \{2,-3,1,-1\}$ since multiplication by zero gives zero, 
but it is non-zero for all other values of $x$ since the product of four non-zero numbers is non-zero. 
