Let's solve the two problems from the image one by one.

Problem 14: Simplify the division

$\frac{2x^2 - 8x}{x^2 - 4} \div \frac{x - 2}{x + 2}$

Step 1: Factorize the numerator and denominator of each expression.

• The numerator of the first fraction, $2x^2 - 8x$, can be factored as: $2x(x - 4)$
• The denominator of the first fraction, $x^2 - 4$, is a difference of squares, so it can be factored as: $(x - 2)(x + 2)$

So the expression becomes:

$\frac{2x(x - 4)}{(x - 2)(x + 2)} \div \frac{x - 2}{x + 2}$

Step 2: Convert the division to multiplication.

When dividing by a fraction, we multiply by its reciprocal:

$\frac{2x(x - 4)}{(x - 2)(x + 2)} \times \frac{x + 2}{x - 2}$

Step 3: Cancel out common factors.

We have $(x + 2)$ in both the numerator and denominator, and $(x - 2)$ in both the numerator and denominator, so we can cancel them:

$\frac{2x(x - 4)}{1}$

Step 4: Simplify the result.

Now, we are left with:

$2x(x - 4)$

Thus, the simplified expression is:

$2x(x - 4)$

Problem 15: Multiply and simplify the expression

$\frac{x^2 + 2x}{x^2 - x - 6} \times \frac{x + 3}{x^2 - 9}$

Step 1: Factorize the expressions.

• The numerator $x^2 + 2x$ can be factored as: $x(x + 2)$
• The denominator $x^2 - x - 6$ can be factored as: $(x - 3)(x + 2)$
• The denominator $x^2 - 9$ is a difference of squares and factors as: $(x - 3)(x + 3)$

So, the expression becomes:

$\frac{x(x + 2)}{(x - 3)(x + 2)} \times \frac{x + 3}{(x - 3)(x + 3)}$

Step 2: Multiply the fractions.

Multiply the numerators and the denominators:

$\frac{x(x + 2)(x + 3)}{(x - 3)(x + 2)(x - 3)(x + 3)}$

Step 3: Cancel out common factors.

We have $(x + 2)$ and $(x + 3)$ in both the numerator and denominator, so we can cancel them:

$\frac{x}{(x - 3)^2}$

Thus, the simplified result is:

$\frac{x}{(x - 3)^2}$

Final Answers:

• Problem 14: The simplified expression is $2x(x - 4)$.
• Problem 15: The simplified expression is $\frac{x}{(x - 3)^2}$.

Would you like further explanation on any of the steps or have additional questions? Here are some related questions for further practice:

1. How would you factorize the expression $x^2 - 9$ using the difference of squares?
2. Can you explain why we canceled the $(x + 2)$ and $(x + 3)$ factors in Problem 15?
3. How would you simplify a fraction with multiple variables in the numerator and denominator?
4. What happens if you try to divide two expressions that do not have any common factors?
5. How do you check your work when simplifying algebraic fractions?

Tip: Always look for common factors in the numerator and denominator before multiplying or dividing expressions to simplify the process.