https://file.aiquickdraw.com/mathbot/file/1727101660023w07njcs0.jpg

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!

Here are 

Solve the following rational expressions: (1) (4n-4)/14, (2) (15x^2 + 30x)/(5x + 10), (3) (4x + 12)/(x^2 + 8x + 15), (4) (x^2 - 9)/(x^2 + 5x + 6), (5) (x^2 - 25)/(4x + 20), (6) (x^2 - 9)/(3 - x).

Learn how to simplify rational expressions step-by-step, including factoring and canceling common factors. Problems covered include (4n-4)/14, (15x^2 + 30x)/(5x + 10), (4x + 12)/(x^2 + 8x + 15), and more. Ideal for students in Grades 9-10.

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