Completely factor the polynomial, if possible.

−8+24x−18x^2

To factor the polynomial \(-8 + 24x - 18x^2\), we can follow these steps:

1. **Arrange the polynomial in standard form**: Write the polynomial with terms in descending order of the exponent.
   \[
   -18x^2 + 24x - 8
   \]

2. **Factor out the greatest common factor (GCF)**: Identify the GCF of the coefficients. In this case, the GCF of \(-18\), \(24\), and \(-8\) is \(2\). Factor out the GCF:
   \[
   2(-9x^2 + 12x - 4)
   \]

3. **Factor the quadratic expression**: Now, we need to factor the quadratic \(-9x^2 + 12x - 4\). We can apply the method of factoring by grouping or using the quadratic formula to check if it factors nicely.

   For factoring by grouping, we first multiply the leading coefficient \(-9\) by the constant term \(-4\), resulting in \(36\).

   We look for two numbers that multiply to \(36\) and add to \(12\). The pair \(6\) and \(6\) satisfies this.

   Rewrite the middle term using these numbers:
   \[
   -9x^2 + 6x + 6x - 4
   \]

   Group the terms:
   \[
   (-9x^2 + 6x) + (6x - 4)
   \]

   Factor each group:
   \[
   3x(-3x + 2) + 2(-3x + 2)
   \]

   Factor out the common binomial:
   \[
   (3x + 2)(-3x + 2)
   \]

4. **Combine with the GCF**: Finally, include the GCF that was factored out earlier:
   \[
   2(3x + 2)(-3x + 2)
   \]

Thus, the completely factored form of the polynomial is:
\[
\boxed{2(3x + 2)(-3x + 2)}
\]

Would you like further details on this process or have any questions?

### Related Questions:
1. How do you determine the greatest common factor (GCF) of a polynomial?
2. What are the steps involved in factoring a quadratic expression?
3. Can the quadratic formula be used to factor polynomials?
4. How do you apply the method of grouping to factor polynomials?
5. How does factoring help in solving polynomial equations?

**Tip:** Always check if a polynomial can be factored further or if it is already in its simplest form.

Completely factor the polynomial, if possible: −8 + 24x − 18x^2

Explore how to completely factor the polynomial −8 + 24x − 18x^2 through step-by-step methods, including factoring by grouping and identifying the greatest common factor (GCF). This problem is ideal for students in Grades 9-11 and covers key algebraic concepts.

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