Using Remainder or Factor Theorems to Find Coefficient I'm studying the remainder and factor theorems and a question asks:

-4 is a root of  $x^4 + ax^3 - 19x^2 - 46x + 120$  What is the value of a?

Since -4 is a root then I can deduce that x+4 is a factor of the polynomial. But that's as far as I get.
Hints?

Let $f(x)=x^4+ax^3-19x^2-46x+120$. We say that $-4$ is a root of $f(x)$ if $f(-4)=0$.
Thus
$$(-4)^4+a(-4)^3-19(-4)^2-46(-4)+120=0.$$
Simplify, and solve for $a$. 
Remark: The problem setter chose the numbers to make  the final answer very simple. 
