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.