By the polynomial remainder theorem, the remainder upon dividing a polynomial [tex]p(x)[/tex] by a linear factor [tex]x-c[/tex] is equal to the value of [tex]p(c)[/tex].
So what you need to do is use synthetic division to compute the quotient and remainder of
[tex]\dfrac{f(m)}{m+4}=\dfrac{-3m^4-17m^3-24m^2-11m+11}{m+4}[/tex]
Synthetic division yields
-4 ... | ... -3 ... -17 ... -24 ... -11 ... 11
... ... | ... ... 12 ... 20 ... 16 ...-20
------------------------------------------------
... ... | ... -3 ..... -5 ..... -4 ...... 5 .... -9
which translates to a quotient of [tex]-3m^3-5m^2-4m+4[/tex] and a remainder term of [tex]-\dfrac{9}{m+4}[/tex]. So [tex]f(-4)=-9[/tex].
