To factorize the quadratic expression \(8 - x - 9x^2\), we will follow a systematic approach. 

First, rewrite the expression in standard form:
\[ -9x^2 - x + 8 \]

Next, we look for two numbers that multiply to \(a \cdot c = (-9) \cdot 8 = -72\) and add up to \(b = -1\).

The numbers that satisfy these conditions are -9 and 8 because:
\[ (-9) \cdot 8 = -72 \]
\[ (-9) + 8 = -1 \]

Using these numbers, we can rewrite the middle term (-x) as a combination of these factors:
\[ -9x^2 - 9x + 8x + 8 \]

Now, group the terms:
\[ (-9x^2 - 9x) + (8x + 8) \]

Factor out the common factors from each group:
\[ -9x(x + 1) + 8(x + 1) \]

Notice that \((x + 1)\) is a common factor:
\[ (-9x + 8)(x + 1) \]

So, the factorized form of the quadratic expression \(8 - x - 9x^2\) is:
\[ (8 - 9x)(x + 1) \]

This gives us the final factorized expression:
\[ 8 - x - 9x^2 = (8 - 9x)(x + 1) \]

Would you like more details or have any questions?

### Suggested Questions:
1. How do you determine which numbers to use for factoring a quadratic expression?
2. What is the grouping method in factoring quadratic expressions?
3. Can you explain how to check if the factorization of a quadratic expression is correct?
4. How do you solve a quadratic equation by factoring?
5. What are some techniques for factoring quadratic expressions that are not easily factorable?

### Tip:
When factoring quadratics, always check your factorization by expanding the factors back out to ensure they produce the original expression. This verifies your work and helps catch any mistakes.

Discover how to factorize the quadratic expression 8 - x - 9x^2 using a detailed step-by-step guide. This problem involves key algebraic concepts such as quadratic equations and factorization theorems. Suitable for students in Grades 9-12.

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