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To factor the quadratic expression \( 44b^2 - 284b - 168 \) completely, let's go through it step-by-step.

### Step 1: Look for a Greatest Common Factor (GCF)
We can see that 44, 284, and 168 all share a common factor of 4, so let's factor out 4:

\[
44b^2 - 284b - 168 = 4(11b^2 - 71b - 42)
\]

Now, we need to factor \( 11b^2 - 71b - 42 \) further.

### Step 2: Factor the Quadratic Expression
Now, we are left with factoring \( 11b^2 - 71b - 42 \).

We need to find two numbers that multiply to \( 11 \times (-42) = -462 \) and add up to \(-71\).

The numbers that satisfy these conditions are \( -77 \) and \( 6 \).

So we can rewrite \( -71b \) as \( -77b + 6b \):

\[
11b^2 - 71b - 42 = 11b^2 - 77b + 6b - 42
\]

### Step 3: Factor by Grouping
Now, we can factor by grouping:

\[
= (11b^2 - 77b) + (6b - 42)
\]

Factor out \( 11b \) from the first group and \( 6 \) from the second group:

\[
= 11b(b - 7) + 6(b - 7)
\]

Now, factor out \( (b - 7) \):

\[
= (b - 7)(11b + 6)
\]

### Step 4: Combine with the GCF
Now we combine everything, including the GCF we factored out earlier:

\[
4(11b^2 - 71b - 42) = 4(b - 7)(11b + 6)
\]

### Final Answer
The completely factored form is:

\[
4(b - 7)(11b + 6)
\]

Would you like further details on any step or have questions?

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Here are some related questions for further practice:

1. How do you factor quadratics when there is a leading coefficient?
2. What is the purpose of factoring by grouping in polynomial expressions?
3. How can you verify that your factorization is correct?
4. What is the role of the Greatest Common Factor in simplifying expressions?
5. How do you handle quadratic expressions that don't have real-number factors?

**Tip:** Always check for a GCF first; it simplifies the expression and makes further factoring easier.

Learn how to factor the quadratic expression 44b^2 - 284b - 168 step-by-step. This solution demonstrates using the Greatest Common Factor (GCF) and the grouping method to simplify quadratic expressions, suitable for high school students in Grades 9-12.

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