We are given the function $g(t) = (t - 1)(t + 4)(t - 2)$.

Finding the $g$-intercept:

The $g$-intercept occurs when $t = 0$. To find it, we substitute $t = 0$ into the function:

$g(0) = (0 - 1)(0 + 4)(0 - 2) = (-1)(4)(-2)$

$g(0) = (-1) \times 4 \times (-2) = 8$

Thus, the $g$-intercept is 8.

Finding the $t$-intercepts:

The $t$-intercepts occur when $g(t) = 0$, which happens when any of the factors in the expression equal zero. So, we set each factor equal to zero:

1. $t - 1 = 0 \implies t = 1$
2. $t + 4 = 0 \implies t = -4$
3. $t - 2 = 0 \implies t = 2$

Thus, the $t$-intercepts are 1, -4, 2.

Would you like more details on how these intercepts were calculated?

Here are some additional related questions for practice:

1. How would you graph the function $g(t)$?
2. What is the behavior of $g(t)$ as $t$ approaches large positive or negative values?
3. How do the signs of the intercepts affect the shape of the graph?
4. What would happen to the intercepts if one of the factors had a different coefficient?
5. Could this be expanded into a cubic function, and how would it look?

Tip: For functions in factored form, the intercepts can be quickly found by solving each factor equal to zero!