sat-dog

Choice B is correct. It's given that $g\left(x\right)=f\left(x-1\right)$. Since $f\left(x\right)=\left(x+6\right)\left(x+5\right)\left(x+1\right)$, it follows that $f\left(x-1\right)=\left(x-1+6\right)\left(x-1+5\right)\left(x-1+1\right)$. Combining like terms yields $f\left(x-1\right)=\left(x+5\right)\left(x+4\right)\left(x\right)$. Therefore, $g\left(x\right)=x\left(x+5\right)\left(x+4\right)$. The x-intercepts of a graph in the xy-plane are the points where $y=0$. The x-coordinates of the x-intercepts of the graph of $y=g\left(x\right)$ in the xy-plane can be found by solving the equation $0=x\left(x+5\right)\left(x+4\right)$. Applying the zero product property to this equation yields three equations: $x=0$, $x+5=0$, and $x+4=0$. Solving each of these equations for $x$ yields $x=0$, $x=-5$, and $x=-4$, respectively. Therefore, the x-intercepts of the graph of $y=g\left(x\right)$ are $\left(0,0\right)$, $\left(-5,0\right)$, and $\left(-4,0\right)$. It follows that the values of $a$, $b$, and $c$ are $0$, $-5$, and $-4$. Thus, the value of $a+b+c$ is $0+\left(-5\right)+\left(-4\right)$, which is equal to $-9$.

Choice A is incorrect. This is the value of $a+b+c$ if $g\left(x\right)=f\left(x+1\right)$.

Choice C is incorrect. This is the value of $a+b+c-1$ if $g\left(x\right)=\left(x-6\right)\left(x-5\right)\left(x-1\right)$.

Choice D is incorrect. This is the value of $a+b+c$ if $f\left(x\right)=\left(x-6\right)\left(x-5\right)\left(x-1\right)$.