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.