Recall the polynomial remainder theorem: if [tex]m-c[/tex] is a factor of some polynomial [tex]p(m)[/tex], then the remainder upon dividing [tex]p(m)[/tex] by [tex]m-c[/tex] is [tex]p(c)[/tex].
Let [tex]c=-1[/tex] and [tex]p(m)=-2m^3+m^2-m+1[/tex]. The remainder left from dividing [tex]p(m)[/tex] by [tex]m+1[/tex] is
[tex]p(-1)=-2(-1)^3+(-1)^2-(-1)+1=5[/tex]
If we subtract 5 from both sides, we'd get a "remainder" of 0, which suggests that we have to add -5 to [tex]p(m)[/tex] to make [tex]m+1[/tex] a factor.
