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