Simplify x2−44x2−12x÷2x+4−x2+5x−6𝑥2−44𝑥2−12𝑥÷2𝑥+4−𝑥2+5𝑥−6.

Let's simplify the given expression step by step. The expression is:

\[
\frac{x^2 - 44x^2 - 12x}{2x + 4 - x^2 + 5x - 6}
\]

### Step 1: Simplify the numerator

The numerator is:

\[
x^2 - 44x^2 - 12x
\]

Combine like terms:

\[
(1 - 44)x^2 - 12x = -43x^2 - 12x
\]

So, the simplified numerator is:

\[
-43x^2 - 12x
\]

### Step 2: Simplify the denominator

The denominator is:

\[
2x + 4 - x^2 + 5x - 6
\]

Combine like terms:

\[
(2x + 5x) + (4 - 6) - x^2 = 7x - x^2 - 2
\]

So, the simplified denominator is:

\[
- x^2 + 7x - 2
\]

### Step 3: Combine the simplified numerator and denominator

Now, we have:

\[
\frac{-43x^2 - 12x}{-x^2 + 7x - 2}
\]

This is the simplified expression.

### Final Simplified Expression:

\[
\frac{-43x^2 - 12x}{-x^2 + 7x - 2}
\]

Would you like further details or have any questions about the simplification? 

Here are 8 related questions that might interest you:
1. How do you factor quadratic expressions like \(x^2 + 5x - 6\)?
2. What are like terms, and how do you combine them in algebraic expressions?
3. Can this simplified expression be factored further?
4. What are the steps to perform polynomial long division?
5. How do you determine the domain of a rational expression?
6. What are common mistakes when simplifying algebraic fractions?
7. How do you check if a rational expression can be simplified further?
8. What is the significance of factoring in simplifying algebraic expressions?

**Tip:** Always double-check your steps when simplifying complex expressions to ensure accuracy.

Learn how to simplify the rational expression (x^2 - 44x^2 - 12x) / (2x + 4 - x^2 + 5x - 6) step-by-step. This problem involves algebraic manipulation of rational expressions and is suitable for high school students (Grades 9-12).

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