If a polynomial p(x) is divisible by q(x), the roots of q(x) are roots of p(x), too.
In this question, p(x)=x³+6x²+kx+12 and q(x)=x+4. Clearly, the root of q(x) is -4. By what was written initially, -4 is a root of p(x). So:
[tex]p(-4)=0\\\\
(-4)^3+6(-4)^2+k(-4)+12=0\\\\
-64+6\cdot16-4k+12=0\\\\
-64+96-4k+12=0\\\\
4k=44\\\\
k=\dfrac{44}{4}\\\\
\boxed{k=11}[/tex]
