To factor the quadratic expression \(3b^2 - 2b - 8\), follow these steps:

1. Multiply the leading coefficient \(3\) and the constant term \(-8\), which gives us \(-24\).
2. Find two numbers that multiply to \(-24\) and add up to \(-2\) (the middle term coefficient). These numbers are \(-6\) and \(4\), since:
   \[
   -6 \times 4 = -24 \quad \text{and} \quad -6 + 4 = -2.
   \]
3. Rewrite the middle term \(-2b\) using \(-6b\) and \(4b\):
   \[
   3b^2 - 6b + 4b - 8.
   \]
4. Factor by grouping:
   \[
   (3b^2 - 6b) + (4b - 8) = 3b(b - 2) + 4(b - 2).
   \]
5. Now factor out the common binomial factor \((b - 2)\):
   \[
   (3b + 4)(b - 2).
   \]

So, the completely factored form is:
\[
(3b + 4)(b - 2).
\]

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you factor quadratic expressions with a leading coefficient other than 1?
2. What if the numbers in step 2 don't exist? What can you do then?
3. Can factoring always be applied to all quadratic equations?
4. What is the difference between factoring by grouping and factoring using the quadratic formula?
5. How do you check if your factorization is correct?

**Tip:** Always double-check your factorization by expanding it back out to ensure it matches the original expression!

Learn how to factor the quadratic expression 3b^2 - 2b - 8 completely using factoring by grouping. This problem includes steps for finding the factors, splitting the middle term, and applying the zero product property. Suitable for students in Grades 8-10.

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