Factor the given expression completely.
2 ab squared minus 8 ab plus 4 ab cubed
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

To factor the expression \( 2ab^2 - 8ab + 4ab^3 \) completely, we can follow these steps:

1. **Identify the common factor:** The greatest common factor (GCF) of the terms is \( 2ab \).

2. **Factor out the GCF:**
   \[
   2ab(b - 4 + 2b^2)
   \]

3. **Rearrange the terms:**
   \[
   2ab(2b^2 + b - 4)
   \]

Now, we can check if the quadratic \( 2b^2 + b - 4 \) can be factored further. To do this, we can use the quadratic formula, or we can look for two numbers that multiply to \( 2 \times -4 = -8 \) and add to \( 1 \) (the coefficient of \( b \)). 

After checking for factor pairs, we find that it cannot be factored nicely with integers. Thus, the completely factored form is:

\[
2ab(2b^2 + b - 4)
\]

So the answer is \( 2ab(2b^2 + b - 4) \).

### Do you want details or have any questions? 

Here are some related questions to consider:
1. How do you apply the quadratic formula to \( 2b^2 + b - 4 \)?
2. What is the significance of finding the GCF when factoring polynomials?
3. Can you provide another example of factoring a polynomial?
4. How would you graph the function represented by the factored expression?
5. What are other methods for factoring polynomials, like grouping or synthetic division?

**Tip:** Always start by factoring out the GCF to simplify the process before tackling more complex factoring methods.

Factor the given expression completely: 2ab squared minus 8 ab plus 4 ab cubed

Discover how to completely factor the expression 2ab^2 - 8ab + 4ab^3, including step-by-step guidance on identifying the GCF and applying the quadratic formula. Ideal for students in Grades 8-10.

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