The image contains problems related to simplifying rational expressions using multiplication or division. Let's work through the problems step-by-step, and I'll provide the simplified forms of the expressions.

a) Simplify: $\frac{x^2 - 9}{x^2 + 6x + 9} \times \frac{y^2 + 4y + 4}{y^2 - 16}$

1. Factor each part:

   • $x^2 - 9 = (x - 3)(x + 3)$
   • $x^2 + 6x + 9 = (x + 3)^2$
   • $y^2 + 4y + 4 = (y + 2)^2$
   • $y^2 - 16 = (y - 4)(y + 4)$

2. Rewrite the expression with the factored form: $\frac{(x - 3)(x + 3)}{(x + 3)^2} \times \frac{(y + 2)^2}{(y - 4)(y + 4)}$

3. Simplify by canceling common factors:

   • Cancel one $x + 3$ term from the numerator and denominator.
   • No other common terms can be canceled.

   The simplified expression is: $\frac{x - 3}{x + 3} \times \frac{(y + 2)^2}{(y - 4)(y + 4)}$ This is the final simplified expression.

b) Simplify: $\frac{3x^2 - 12}{15x} \div \frac{4x^2 - 8x}{2x^2}$

1. Factor each part:

   • $3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2)$
   • $4x^2 - 8x = 4x(x - 2)$

2. Rewriting the division as multiplication of the reciprocal: $\frac{3(x - 2)(x + 2)}{15x} \times \frac{2x^2}{4x(x - 2)}$

3. Simplify:

   • Cancel the $x - 2$ from both the numerator and denominator.
   • Simplify $\frac{3}{15} = \frac{1}{5}$ and $\frac{2x^2}{4x} = \frac{x}{2}$.

   The final expression becomes: $\frac{(x + 2)}{5} \times \frac{x}{2} = \frac{x(x + 2)}{10}$ This is the simplified result.

For the rest of the questions, let me know if you'd like me to proceed similarly or offer additional guidance. Would you like detailed steps for parts c, d, and e?

Related Questions:

1. How do you identify factors of quadratic expressions quickly?
2. What is the procedure for multiplying two rational expressions?
3. How do you convert division of rational expressions into multiplication?
4. When simplifying, what common factors should be prioritized for cancellation?
5. How can factoring the difference of squares help in simplification?

Tip:

Always factor each part of the rational expressions completely before simplifying. This makes the cancellation of common factors much easier.