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The trinomial provided is \(-8x^2 - 12x + 36\). Let's factor it step by step.

First, we'll factor out the greatest common factor, which is \(-4\) in this case:
\[
-4(2x^2 + 3x - 9)
\]

Next, we need to factor the quadratic \(2x^2 + 3x - 9\). We will look for two numbers that multiply to \(2 \times (-9) = -18\) and add to \(3\). These numbers are \(6\) and \(-3\).

Rewriting the middle term using these numbers:
\[
2x^2 + 6x - 3x - 9
\]

Now, we group and factor by grouping:
\[
(2x^2 + 6x) - (3x + 9)
\]
\[
2x(x + 3) - 3(x + 3)
\]

Factoring out the common factor \((x + 3)\):
\[
(2x - 3)(x + 3)
\]

Therefore, the factored form of the original trinomial is:
\[
-4(2x - 3)(x + 3)
\]

Do you need any further details on this or have any other questions about factoring?

Here are five follow-up questions that can expand on this topic:
1. What is the quadratic formula and how can it be used to check the roots of the equation \(2x^2 + 3x - 9 = 0\)?
2. Can you solve for \(x\) in the equation \(-8x^2 - 12x + 36 = 0\) using this factorization?
3. What are the general steps for factoring a quadratic expression?
4. How can one determine the greatest common factor of a polynomial?
5. Can you explain the difference between factoring by grouping and other factoring techniques?

**Tip:** When factoring polynomials, always check if there is a greatest common factor you can pull out first. This simplifies the rest of the factoring process.

Learn how to factor the trinomial -8x^2 - 12x + 36 with a detailed step-by-step approach. This guide covers key algebraic concepts such as factoring by grouping, identifying the greatest common factor, and simplifying expressions. Suitable for students in Grades 9-10.

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