To find the 7th term of the geometric sequence, we first need to identify the common ratio, $r$, of the sequence.

Given sequence: $6x^2, -24x^6, 96x^{10}, \dots$

Step 1: Determine the common ratio $r$

The common ratio $r$ is found by dividing the second term by the first term:

$r = \frac{-24x^6}{6x^2} = \frac{-24}{6} \times \frac{x^6}{x^2} = -4x^4$

Step 2: Verify the common ratio with subsequent terms

Now, let's verify $r$ by dividing the third term by the second term:

$r = \frac{96x^{10}}{-24x^6} = \frac{96}{-24} \times \frac{x^{10}}{x^6} = -4x^4$

Since the common ratio is consistent, we can use it to find the 7th term.

Step 3: Use the formula for the $n$-th term of a geometric sequence

The formula for the $n$-th term of a geometric sequence is:

$a_n = a_1 \times r^{n-1}$

For the 7th term $a_7$:

$a_7 = 6x^2 \times (-4x^4)^{7-1} = 6x^2 \times (-4x^4)^6$

Step 4: Simplify the expression

$(-4x^4)^6 = (-4)^6 \times (x^4)^6 = 4096x^{24}$

Thus,

$a_7 = 6x^2 \times 4096x^{24} = 24576x^{26}$

Final Answer

The 7th term of the geometric sequence is $\boxed{24576x^{26}}$.

Would you like a more detailed explanation, or do you have any questions? Here are some related questions:

1. How do you find the common ratio in a geometric sequence?
2. What is the formula for the general term of a geometric sequence?
3. How can you verify if a sequence is geometric?
4. What is the 5th term of the same sequence?
5. How does changing the common ratio affect the sequence?

Tip: In a geometric sequence, the ratio between consecutive terms remains constant, which is key to solving problems like this one.