The function f is defined by f x = x - 6 x - 2 x

The correct answer is $76$. It's given that the graph of $y=g\left(x\right)$ is the result of translating the graph of $y=f\left(x\right)$ up $4$ units in the xy-plane. It follows that the graph of $y=g\left(x\right)$ is the same as the graph of $y=f\left(x\right)+4$. Substituting $g\left(x\right)$ for $y$ in the equation $y=f\left(x\right)+4$ yields $g\left(x\right)=f\left(x\right)+4$. It's given that $f\left(x\right)=\left(x-6\right)\left(x-2\right)\left(x+6\right)$. Substituting $\left(x-6\right)\left(x-2\right)\left(x+6\right)$ for $f\left(x\right)$ in the equation $g\left(x\right)=f\left(x\right)+4$ yields $g\left(x\right)=\left(x-6\right)\left(x-2\right)\left(x+6\right)+4$. Substituting $0$ for $x$ in this equation yields $g\left(0\right)=\left(0-6\right)\left(0-2\right)\left(0+6\right)+4$, or $g\left(0\right)=76$. Thus, the value of $g\left(0\right)$ is $76$.