Let's solve each question step by step to identify the correct simplified expressions:

Problem 3

The given function is:

$f(x) = \frac{3x}{x^2 - 9}$

1. Factorize the denominator: $x^2 - 9 = (x - 3)(x + 3)$

   The function becomes: $f(x) = \frac{3x}{(x - 3)(x + 3)}$

2. Check for common factors: There are no common factors between $3x$ and $(x - 3)(x + 3)$, so the function cannot be simplified further.

   Answer: "Does not simplify."

Problem 4

The given function is:

$g(x) = \frac{2x^2 - 4x}{10x^2 + 6x}$

1. Factorize numerator and denominator:

   • Numerator: $2x^2 - 4x = 2x(x - 2)$
   • Denominator: $10x^2 + 6x = 2x(5x + 3)$

   The function becomes: $g(x) = \frac{2x(x - 2)}{2x(5x + 3)}$

2. Cancel the common factor $2x$: $g(x) = \frac{x - 2}{5x + 3}$

   Answer: $\frac{x - 2}{5x + 3}$

Problem 5

The given function is:

$h(x) = \frac{4x^2 + 10x}{7x^2}$

1. Factorize numerator:

   • Numerator: $4x^2 + 10x = 2x(2x + 5)$

   The function becomes: $h(x) = \frac{2x(2x + 5)}{7x^2}$

2. Cancel the common factor $x$ (assuming $x \neq 0$): $h(x) = \frac{2(2x + 5)}{7x}$

   Answer: $\frac{2(2x + 5)}{7x}$

Final Answers:

1. Problem 3: Does not simplify.
2. Problem 4: $\frac{x - 2}{5x + 3}$
3. Problem 5: $\frac{2(2x + 5)}{7x}$

Would you like a detailed breakdown for any specific part or additional explanation? Here are some related questions for practice:

1. How do you identify common factors between numerator and denominator?
2. What are the common errors in simplifying rational expressions?
3. Can you simplify $\frac{x^2 - 4}{x^2 + 4x + 4}$?
4. How do you handle restrictions on $x$ when simplifying expressions?
5. How do you approach a rational function that contains higher powers of $x$?

Tip: Always factorize completely before canceling terms to ensure you simplify correctly!