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We are tasked with factoring the quadratic expression \( 8b^2 - 51b + 18 \). Let's go through the factorization step by step and then check the provided answer choices.

### Expression:
\[ 8b^2 - 51b + 18 \]

#### Step 1: Multiply \( a \) and \( c \).
Here, the coefficients are:
- \( a = 8 \)
- \( b = -51 \)
- \( c = 18 \)

Multiply \( a \) and \( c \):
\[ 8 \times 18 = 144 \]

#### Step 2: Find two numbers that multiply to 144 and add to -51.
We need two numbers whose product is 144 and sum is -51. These numbers are:
- \( -3 \) and \( -48 \) (because \( -3 \times -48 = 144 \) and \( -3 + -48 = -51 \)).

#### Step 3: Rewrite the middle term using these two numbers.
We can now rewrite \( -51b \) as \( -3b - 48b \):
\[
8b^2 - 3b - 48b + 18
\]

#### Step 4: Factor by grouping.
Now, we factor the expression by grouping the terms:
\[
(8b^2 - 3b) - (48b - 18)
\]

Factor out the common factors from each group:
\[
b(8b - 3) - 6(8b - 3)
\]

Now, factor out the common binomial factor \( (8b - 3) \):
\[
(b - 6)(8b - 3)
\]

#### Step 5: Compare with the answer choices.
The correct factorization is:
\[
(b - 6)(8b - 3)
\]
This matches option **D**.

### Final Answer:
The correct answer is **D: \( (b - 6)(8b - 3) \)**.

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Would you like more details on the steps, or do you have any other questions?

Here are 5 related questions to consider:
1. How do you know when to use the method of grouping in factoring?
2. Can all quadratic expressions be factored using simple integer solutions?
3. What is the role of the discriminant in factoring quadratics?
4. How can you verify if a factorization is correct?
5. How would the approach differ if \( b^2 - 4ac \) were negative?

**Tip:** Always check your factorization by expanding it back to ensure you get the original quadratic equation.

Factor the quadratic expression 8b² - 51b + 18.

Learn how to factor the quadratic expression 8b² - 51b + 18 step-by-step. This solution covers key algebraic concepts, including factoring by grouping. Suitable for high school students.

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