The expression provided in the image is:

$\frac{x^2 - 5x - 24}{6x + 2x^2} \cdot \frac{5x^2}{8 - x}$

Let's simplify it step by step:

Step 1: Factorize where possible

• The numerator $x^2 - 5x - 24$ can be factored.
• The denominator $6x + 2x^2$ can also be factored.

Factorization of $x^2 - 5x - 24$:

We look for two numbers that multiply to $-24$ and add to $-5$. These numbers are $-8$ and $3$. Thus, the factorization is: $x^2 - 5x - 24 = (x - 8)(x + 3)$

Factorization of $6x + 2x^2$:

Factor out the common term $2x$: $6x + 2x^2 = 2x(x + 3)$

Step 2: Rewrite the expression

Now we rewrite the expression with the factored terms: $\frac{(x - 8)(x + 3)}{2x(x + 3)} \cdot \frac{5x^2}{8 - x}$

Step 3: Simplify the expression

We can cancel out common terms:

• The $x + 3$ in the numerator and denominator cancels out.
• We are left with: $\frac{(x - 8)}{2x} \cdot \frac{5x^2}{8 - x}$

Notice that $x - 8$ and $8 - x$ are negatives of each other: $x - 8 = -(8 - x)$

Thus, the expression simplifies to: $-\frac{5x}{2}$

This is the simplified result.

Would you like more details on any step or further clarification? Here are a few related questions:

1. How do you factor quadratic expressions?
2. What is the significance of canceling terms in rational expressions?
3. How can you simplify expressions with negative terms?
4. What are the common mistakes to avoid when canceling terms in fractions?
5. How do you solve a quadratic equation by factoring?

Tip: Always check if terms can be factored to simplify expressions before multiplying or dividing them.