https://file.aiquickdraw.com/mathbot/file/1727328763867oiwm9h2b.jpg

The expression in the image is:

\[
8t^3 - 4t^2 - 40t
\]

### Steps to factor the expression completely:

1. **Factor out the greatest common factor (GCF):**
   - The GCF of \(8t^3\), \(-4t^2\), and \(-40t\) is \(4t\).
   - Factor \(4t\) from the expression:
   
   \[
   8t^3 - 4t^2 - 40t = 4t(2t^2 - t - 10)
   \]

2. **Factor the quadratic expression \(2t^2 - t - 10\):**
   - To factor \(2t^2 - t - 10\), we look for two numbers that multiply to \(2 \times (-10) = -20\) and add up to \(-1\). These numbers are \(4\) and \(-5\).
   - Rewrite the middle term:
   
   \[
   2t^2 - 5t + 4t - 10
   \]
   
3. **Group the terms:**
   
   \[
   (2t^2 - 5t) + (4t - 10)
   \]

4. **Factor each group:**
   
   \[
   t(2t - 5) + 2(2t - 5)
   \]

5. **Factor out the common binomial factor:**
   
   \[
   (t + 2)(2t - 5)
   \]

6. **Final factored form:**
   
   \[
   4t(t + 2)(2t - 5)
   \]

### Final Answer:
\[
4t(t + 2)(2t - 5)
\]

Would you like more details on any of the steps? 

Here are some related questions to explore:

1. How do you find the greatest common factor (GCF) of terms in an expression?
2. How do you factor quadratic trinomials?
3. What is the importance of grouping in factoring?
4. How do you check if your factorization is correct?
5. What other methods can be used to factor polynomials?

**Tip:** Always factor out the GCF first before attempting other factoring techniques for a cleaner, more straightforward approach!

Factor the expression completely: 8t^3 - 4t^2 - 40t

Learn how to factor the expression 8t^3 - 4t^2 - 40t completely. This step-by-step guide covers key algebraic concepts such as the Greatest Common Factor (GCF) and quadratic factoring techniques. Suitable for students in Grades 8-10.

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