Let's solve the questions one by one from the image you've provided. The task is to simplify the rational algebraic expressions.

1. $\frac{45}{10x - 10}$

First, factor the denominator: $10x - 10 = 10(x - 1)$ So, the expression becomes: $\frac{45}{10(x - 1)} = \frac{45}{10} \cdot \frac{1}{x - 1} = \frac{9}{2(x - 1)}$ Answer: $\frac{9}{2(x - 1)}$

2. $\frac{16p}{28p}$

Simplify by canceling the common factor $p$: $\frac{16p}{28p} = \frac{16}{28} = \frac{4}{7}$ Answer: $\frac{4}{7}$

3. $\frac{28}{7x - 4}$

No common factors, so we leave it as is.
Answer: $\frac{28}{7x - 4}$

4. $\frac{x - 4}{3x - 12x}$

Simplify the denominator: $3x - 12x = -9x$ The expression becomes: $\frac{x - 4}{-9x}$ Answer: $\frac{x - 4}{-9x}$

5. $\frac{x + 6}{x^2 + 5x - 6}$

Factor the denominator: $x^2 + 5x - 6 = (x - 1)(x + 6)$ Now simplify the expression: $\frac{x + 6}{(x - 1)(x + 6)} = \frac{1}{x - 1}$ Answer: $\frac{1}{x - 1}$

6. $\frac{b^2 - 49}{b^2 + 3b - 28}$

Factor both numerator and denominator: $b^2 - 49 = (b - 7)(b + 7)$ $b^2 + 3b - 28 = (b - 4)(b + 7)$ Now simplify: $\frac{(b - 7)(b + 7)}{(b - 4)(b + 7)} = \frac{b - 7}{b - 4}$ Answer: $\frac{b - 7}{b - 4}$

7. $\frac{2x^2 - 20x + 42}{2x^2 - 8x}$

Factor both numerator and denominator: $2x^2 - 20x + 42 = 2(x^2 - 10x + 21) = 2(x - 3)(x - 7)$ $2x^2 - 8x = 2x(x - 4)$ Now simplify: $\frac{2(x - 3)(x - 7)}{2x(x - 4)} = \frac{(x - 3)(x - 7)}{x(x - 4)}$ Answer: $\frac{(x - 3)(x - 7)}{x(x - 4)}$

8. $\frac{41 - 4}{60 - 20}$

Simplify the constants: $41 - 4 = 37, \quad 60 - 20 = 40$ So, the expression becomes: $\frac{37}{40}$ Answer: $\frac{37}{40}$

9. $\frac{3r^2 - 39r + 90}{r^2 - 3r - 70}$

Factor both numerator and denominator: $3r^2 - 39r + 90 = 3(r^2 - 13r + 30) = 3(r - 3)(r - 10)$ $r^2 - 3r - 70 = (r - 10)(r + 7)$ Now simplify: $\frac{3(r - 3)(r - 10)}{(r - 10)(r + 7)} = \frac{3(r - 3)}{r + 7}$ Answer: $\frac{3(r - 3)}{r + 7}$

10. $\frac{2x^2 + 2x - 80}{2x^2 - 24x + 64}$

Factor both numerator and denominator: $2x^2 + 2x - 80 = 2(x^2 + x - 40) = 2(x - 5)(x + 8)$ $2x^2 - 24x + 64 = 2(x^2 - 12x + 32) = 2(x - 4)(x - 8)$ Now simplify: $\frac{2(x - 5)(x + 8)}{2(x - 4)(x - 8)} = \frac{(x - 5)(x + 8)}{(x - 4)(x - 8)}$ Answer: $\frac{(x - 5)(x + 8)}{(x - 4)(x - 8)}$

Do you want details on any specific steps or have any further questions?

Here are 5 related questions to expand on this:

1. How can factoring help in simplifying rational expressions?
2. What are the common methods to factor quadratic expressions?
3. How do you identify common factors between the numerator and denominator?
4. Can all rational expressions be simplified by factoring?
5. How do you deal with rational expressions that include negative exponents?

Tip: Always check for common factors before simplifying any rational expression; this will save time and reduce errors.