To factor the quadratic expression \(-x^2 + 11x - 18\), follow these steps:

### Step 1: Factor out the negative sign
First, factor out the negative sign from the entire expression:

\[
-x^2 + 11x - 18 = -(x^2 - 11x + 18)
\]

### Step 2: Factor the quadratic expression \(x^2 - 11x + 18\)
We need to find two numbers that multiply to \(18\) (the constant term) and add to \(-11\) (the coefficient of the linear term).

The numbers that work are \(-9\) and \(-2\), because:

\[
-9 \times -2 = 18 \quad \text{and} \quad -9 + -2 = -11
\]

So, the factorization is:

\[
x^2 - 11x + 18 = (x - 9)(x - 2)
\]

### Step 3: Apply the negative sign back
Now, include the negative sign we factored out earlier:

\[
-x^2 + 11x - 18 = -(x - 9)(x - 2)
\]

### Final Answer:
\[
-x^2 + 11x - 18 = -(x - 9)(x - 2)
\]

Would you like more details, or do you have any other questions?

### Follow-Up Questions:
1. How do you determine which numbers to use when factoring a quadratic expression?
2. What is the significance of factoring out a negative sign first?
3. How can you check if the factorization of a quadratic is correct?
4. What happens if the quadratic cannot be factored over the integers?
5. How would you factor a quadratic with a leading coefficient other than 1?

**Tip:** When factoring, always consider if there's a common factor to factor out first, including a negative sign if the leading coefficient is negative.

Learn how to factorize the quadratic expression -x^2 + 11x - 18 step-by-step. This problem covers key algebraic concepts, including factoring quadratic equations and handling negative signs in factorization. Suitable for high school students in Grades 9-12.

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