Let's go step by step and solve each of the rational expressions in the image.

1. $\frac{4n - 4}{14}$

Solution: Factor out the common term in the numerator: $\frac{4n - 4}{14} = \frac{4(n - 1)}{14}$ Now simplify by dividing both the numerator and denominator by 2: $\frac{4(n - 1)}{14} = \frac{2(n - 1)}{7}$ So, the simplified expression is: $\frac{2(n - 1)}{7}$

2. $\frac{15x^2 + 30x}{5x + 10}$

Solution: Factor out the greatest common factor (GCF) in both the numerator and denominator: $\frac{15x^2 + 30x}{5x + 10} = \frac{15x(x + 2)}{5(x + 2)}$ Cancel out the common factor $(x + 2)$: $\frac{15x(x + 2)}{5(x + 2)} = \frac{15x}{5} = 3x$ So, the simplified expression is: $3x$

3. $\frac{4x + 12}{x^2 + 8x + 15}$

Solution: Factor both the numerator and the denominator:

• Numerator: $4x + 12 = 4(x + 3)$
• Denominator: $x^2 + 8x + 15 = (x + 3)(x + 5)$

Now cancel out the common factor $(x + 3)$: $\frac{4(x + 3)}{(x + 3)(x + 5)} = \frac{4}{x + 5}$ So, the simplified expression is: $\frac{4}{x + 5}$

4. $\frac{x^2 - 9}{x^2 + 5x + 6}$

Solution: Factor both the numerator and the denominator:

• Numerator: $x^2 - 9 = (x - 3)(x + 3)$ (difference of squares)
• Denominator: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Now cancel out the common factor $(x + 3)$: $\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}$ So, the simplified expression is: $\frac{x - 3}{x + 2}$

5. $\frac{x^2 - 25}{4x + 20}$

Solution: Factor both the numerator and the denominator:

• Numerator: $x^2 - 25 = (x - 5)(x + 5)$ (difference of squares)
• Denominator: $4x + 20 = 4(x + 5)$

Now cancel out the common factor $(x + 5)$: $\frac{(x - 5)(x + 5)}{4(x + 5)} = \frac{x - 5}{4}$ So, the simplified expression is: $\frac{x - 5}{4}$

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

Solution: Factor the numerator:

• Numerator: $x^2 - 9 = (x - 3)(x + 3)$ (difference of squares)

Notice that the denominator $3 - x$ can be rewritten as $-(x - 3)$, since: $3 - x = -(x - 3)$ So, the expression becomes: $\frac{(x - 3)(x + 3)}{-(x - 3)} = \frac{-(x + 3)}{1} = -(x + 3)$ Thus, the simplified expression is: $-(x + 3)$

Let me know if you'd like any further details on these steps or have any questions!

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