The function g is defined by g x = x + 14 t - x ,

The correct answer is $336$. By the zero product property, if $\left(x+14\right)\left(t-x\right)=0$, then $x+14=0$, which gives $x=-14$, or $\left(t-x\right)=0$, which gives $x=t$. Therefore, $g\left(x\right)=0$ when $x=-14$ and when $x=t$. Since the graph of $y=g\left(x\right)$ passes through the point $\left(24,0\right)$, it follows that $g\left(24\right)=0$, so $t=24$. Substituting $24$ for $t$ in the equation $g\left(x\right)=\left(x+14\right)\left(t-x\right)$ yields $g\left(x\right)=\left(x+14\right)\left(24-x\right)$. The value of $g\left(0\right)$ can be calculated by substituting $0$ for $x$ in this equation, which yields $g\left(0\right)=\left(0+14\right)\left(24-0\right)$, or $g\left(0\right)=336$.