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Let's factor the quadratic equation \( 2x^2 + 11x + 12 \).

We are looking to express it in the form:

\[
(2x + \_)(x + \_)
\]

### Steps:

1. Multiply the coefficient of \(x^2\) (which is 2) by the constant term (which is 12). This gives \( 2 \times 12 = 24 \).
2. Now, we need two numbers that multiply to give 24 and add to give the middle coefficient (11). The numbers are 3 and 8 because \( 3 \times 8 = 24 \) and \( 3 + 8 = 11 \).
3. Rewrite the middle term \( 11x \) using 3 and 8:
   \[
   2x^2 + 3x + 8x + 12
   \]
4. Group the terms and factor by grouping:
   \[
   (2x^2 + 3x) + (8x + 12)
   \]
   \[
   x(2x + 3) + 4(2x + 3)
   \]
5. Factor out the common binomial:
   \[
   (2x + 3)(x + 4)
   \]

So, the factored form is:

\[
(2x + 3)(x + 4)
\]

If you have any further questions or need more details, feel free to ask!

Here are 5 related questions:
1. How would you factor a quadratic if the leading coefficient is negative?
2. Can you factor a quadratic where the middle term is missing (e.g., \(2x^2 + 12\))?
3. How do you solve a quadratic equation once it’s factored?
4. What happens if no two numbers add up to the middle term when factoring?
5. How can you verify if your factorization is correct?

**Tip**: To verify your factorization, expand the factors back to the original quadratic equation.

Learn how to factor the quadratic equation 2x^2 + 11x + 12 step-by-step using the grouping method. This solution covers key algebraic concepts suitable for students in Grades 9-10.

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