The Remainder Theorem tells us that the remainder of a polynomial $p(x)$ when divided by a linear divisor $x - c$ is simply $p(c)$.

In your case, the polynomial $p(x) = 3x^4 - 6x^2 - 8x - 2$ is being divided by $g(x) = x - 2$. According to the Remainder Theorem, the remainder is $p(2)$, because $g(x) = x - 2$ implies $c = 2$.

Now, we can calculate $p(2)$:

$p(x) = 3x^4 - 6x^2 - 8x - 2$

Substitute $x = 2$ into the polynomial:

$p(2) = 3(2)^4 - 6(2)^2 - 8(2) - 2$

Simplifying:

$p(2) = 3(16) - 6(4) - 8(2) - 2$ $p(2) = 48 - 24 - 16 - 2$ $p(2) = 6$

So, the remainder when $p(x)$ is divided by $g(x) = x - 2$ is 6.

---

Would you like more details on the Remainder Theorem, or do you have any other questions? Here are a few related questions you could explore:

1. How does the Factor Theorem relate to the Remainder Theorem?
2. How would you find the remainder if $g(x)$ was $x + 3$?
3. What is synthetic division, and how can it be used to find remainders?
4. How can the Remainder Theorem be applied in solving polynomial equations?
5. What happens when the remainder is zero in polynomial division?

Tip: The Remainder Theorem is a quick way to find remainders for linear divisors. If the divisor is not linear, you’ll need to use long division or synthetic division instead.