Answer
$x+4$ is a factor of the polynomial.
Work Step by Step
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and if $(x-c)$ is a factor of $f(x)$, then $f(c)=0$.
We check if $x+4$ is a factor by testing if $-4$ is a zero:
$P(-4)=(-4)^5+4(-4)^4-7(-4)^3-23(-4)^3+23(-4)+12=0$
Since we got $0$, we know that $x+4$ is indeed a factor of the polynomial.
