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The problem requires you to use the product rule to calculate the derivative of the function \( f(t) \times g(t) \), where the functions are:

\[
f(t) = (3t^2 + t - 21)
\]
\[
g(t) = (t^4 + t^2 - 2)
\]

### Product Rule

The product rule states that the derivative of the product of two functions \( f(t) \) and \( g(t) \) is given by:

\[
\frac{d}{dt}[f(t) \cdot g(t)] = f'(t) \cdot g(t) + f(t) \cdot g'(t)
\]

### Step 1: Calculate \( f'(t) \) and \( g'(t) \)

- \( f(t) = 3t^2 + t - 21 \)
  - \( f'(t) = 6t + 1 \)

- \( g(t) = t^4 + t^2 - 2 \)
  - \( g'(t) = 4t^3 + 2t \)

### Step 2: Apply the Product Rule

Now substitute \( f(t) \), \( f'(t) \), \( g(t) \), and \( g'(t) \) into the product rule formula:

\[
\frac{d}{dt}[f(t) \cdot g(t)] = (6t + 1) \cdot (t^4 + t^2 - 2) + (3t^2 + t - 21) \cdot (4t^3 + 2t)
\]

### Step 3: Expand and Simplify

1. Expand \( (6t + 1) \cdot (t^4 + t^2 - 2) \):
   \[
   6t \cdot t^4 + 6t \cdot t^2 + 6t \cdot (-2) + 1 \cdot t^4 + 1 \cdot t^2 + 1 \cdot (-2) = 6t^5 + 6t^3 - 12t + t^4 + t^2 - 2
   \]
   So, \( (6t + 1) \cdot (t^4 + t^2 - 2) = 6t^5 + t^4 + 6t^3 + t^2 - 12t - 2 \).

2. Expand \( (3t^2 + t - 21) \cdot (4t^3 + 2t) \):
   \[
   3t^2 \cdot 4t^3 + 3t^2 \cdot 2t + t \cdot 4t^3 + t \cdot 2t + (-21) \cdot 4t^3 + (-21) \cdot 2t = 12t^5 + 6t^3 + 4t^4 + 2t^2 - 84t^3 - 42t
   \]
   So, \( (3t^2 + t - 21) \cdot (4t^3 + 2t) = 12t^5 + 4t^4 - 78t^3 + 2t^2 - 42t \).

### Step 4: Combine Like Terms

Finally, add the two expanded expressions:

\[
(6t^5 + t^4 + 6t^3 + t^2 - 12t - 2) + (12t^5 + 4t^4 - 78t^3 + 2t^2 - 42t) = 18t^5 + 5t^4 - 72t^3 + 3t^2 - 54t - 2
\]

### Final Answer:

The derivative of the function using the product rule is:

\[
\boxed{18t^5 + 5t^4 - 72t^3 + 3t^2 - 54t - 2}
\]

Let me know if you need any more details or have any questions!

### Follow-up Questions:
1. What is the product rule and how is it used to differentiate functions?
2. How do you calculate the derivative of a polynomial function?
3. What are some common mistakes when applying the product rule?
4. How would you find the second derivative of the given function?
5. How does the product rule relate to the chain rule in differentiation?

### Tip:
Always double-check your calculations for sign errors or missed terms, especially when expanding and combining like terms!

Use the product rule to calculate the derivative of the function f(t) = (3t^2 + t - 21)(t^4 + t^2 - 2)

Learn how to differentiate the product of two functions f(t) = (3t^2 + t - 21)(t^4 + t^2 - 2) using the product rule. This problem is suitable for high school students learning Calculus and covers key concepts like polynomial differentiation.

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