To insert 4 geometric means between $-2$ and $486$, we first need to understand that a geometric sequence has a common ratio, $r$, between consecutive terms.

Let the five terms of the geometric sequence be:

$-2, G_1, G_2, G_3, G_4, 486$

In a geometric sequence, the ratio between consecutive terms is constant. The general form of a geometric sequence is:

$a, ar, ar^2, ar^3, \dots, ar^n$

Here, $-2$ is the first term, and $486$ is the sixth term (after inserting 4 geometric means). We can express the sixth term as:

$486 = -2 \times r^5$

Now, solve for $r$:

$r^5 = \frac{486}{-2} = -243$

Taking the fifth root of both sides:

$r = \sqrt[5]{-243} = -3$

So, the common ratio is $r = -3$.

Now we can calculate the geometric means:

1. First term: $-2$
2. Second term: $-2 \times (-3) = 6$
3. Third term: $6 \times (-3) = -18$
4. Fourth term: $-18 \times (-3) = 54$
5. Fifth term: $54 \times (-3) = -162$
6. Sixth term: $-162 \times (-3) = 486$

Thus, the 4 geometric means between $-2$ and $486$ are:

$6, -18, 54, -162$

Would you like more details or further clarification?

Here are 5 related questions to expand your understanding:

1. How do you find the common ratio in any geometric sequence?
2. What are the differences between arithmetic and geometric sequences?
3. How would the method change if we needed to insert arithmetic means instead of geometric means?
4. Can a geometric sequence contain zero as one of its terms? Why or why not?
5. What happens to the common ratio if the terms in a geometric sequence alternate between positive and negative values?

Tip: The sign of the common ratio determines whether the sequence alternates between positive and negative terms!