https://file.aiquickdraw.com/mathbot/file/17284800168002hx3y3s6.jpg

We are asked to factor the quadratic expression:

\[
2x^2 - 9x - 18
\]

The given hint suggests that we need to express this in the form:

\[
(2x + \text{?})(x - \text{?})
\]

### Step-by-Step Solution:

1. **Identify the form**: The quadratic is in the form \(ax^2 + bx + c\), where \(a = 2\), \(b = -9\), and \(c = -18\).

2. **Multiply \(a\) and \(c\)**: Multiply the coefficient of \(x^2\) (which is 2) by the constant term (which is -18):
   \[
   2 \times -18 = -36
   \]

3. **Find factors of -36**: We need to find two numbers that multiply to give -36 and add up to -9 (the middle term's coefficient). These numbers are:
   \[
   -12 \text{ and } 3 \quad (\text{since } -12 \times 3 = -36 \text{ and } -12 + 3 = -9)
   \]

4. **Rewrite the middle term**: Rewrite the middle term \(-9x\) as \(-12x + 3x\):
   \[
   2x^2 - 12x + 3x - 18
   \]

5. **Factor by grouping**: Group the terms:
   \[
   (2x^2 - 12x) + (3x - 18)
   \]
   Factor out the common factors from each group:
   \[
   2x(x - 6) + 3(x - 6)
   \]

6. **Factor out the common binomial**: Both terms contain \((x - 6)\), so factor that out:
   \[
   (2x + 3)(x - 6)
   \]

Thus, the fully factored form is:

\[
(2x + 3)(x - 6)
\]

### Final Answer:
- The value for the green box (first missing term) is **3**.
- The value for the grey box (second missing term) is **6**.

Would you like more details on any of the steps? 

Here are some related questions you can explore:
1. How do you factor quadratic expressions using the AC method?
2. Can all quadratic expressions be factored?
3. What is the difference between factoring and solving a quadratic equation?
4. How do you verify that a factorization is correct?
5. How would you factor quadratics with coefficients greater than 2?

**Tip**: Always check your factorization by expanding the factors back to ensure they result in the original quadratic expression.

Factor the quadratic expression: 2x^2 - 9x - 18.

Learn how to factor the quadratic expression 2x^2 - 9x - 18 using the AC method. The problem demonstrates key algebraic concepts such as factoring by grouping. Suitable for students in Grades 8-10.

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