Simplify: $$ \frac{3 x^{2}+17 x+10}{3 x^{2}+32 x+20} $$ $$ \frac{3 x+2}{3 x+4} $$ $$ \frac{1}{2} $$ $$ \frac{x+5}{x+10} $$ $$ \frac{17 x+10}{32 x+20} $$

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Let's simplify each of the given fractions:

1. **Simplify** $$ \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} $$

**Step 1:** Factor the numerator and the denominator.
   
   - **Numerator:** $$ 3x^{2} + 17x + 10 $$
     $$
     3x^{2} + 17x + 10 = (3x + 2)(x + 5)
     $$
   
   - **Denominator:** $$ 3x^{2} + 32x + 20 $$
     $$
     3x^{2} + 32x + 20 = (3x + 2)(x + 10)
     $$
   
   **Step 2:** Cancel the common factor $$(3x + 2)$$.
   $$
   \frac{(3x + 2)(x + 5)}{(3x + 2)(x + 10)} = \frac{x + 5}{x + 10}
   $$
   
   **Simplified Form:**
   $$
   \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} = \frac{x + 5}{x + 10}
   $$

2. **Simplify** $$ \frac{3x + 2}{3x + 4} $$

This fraction cannot be simplified further because the numerator and the denominator do not have any common factors other than 1.

**Simplified Form:**
   $$
   \frac{3x + 2}{3x + 4}
   $$

3. **Simplify** $$ \frac{1}{2} $$

This fraction is already in its simplest form.

**Simplified Form:**
   $$
   \frac{1}{2}
   $$

4. **Simplify** $$ \frac{x + 5}{x + 10} $$

This fraction cannot be simplified further because the numerator and the denominator do not have any common factors other than 1.

**Simplified Form:**
   $$
   \frac{x + 5}{x + 10}
   $$

5. **Simplify** $$ \frac{17x + 10}{32x + 20} $$

**Step 1:** Factor the denominator.
   $$
   32x + 20 = 4(8x + 5)
   $$
   
   **Step 2:** Check if the numerator can be factored further or if there's a common factor with the denominator. 
   
   The numerator $$17x + 10$$ does not have any common factors with the denominator $$8x + 5$$ besides 1.
   
   **Simplified Form:**
   $$
   \frac{17x + 10}{32x + 20} = \frac{17x + 10}{4(8x + 5)}
   $$

**Summary of Simplified Fractions:**

1. $$ \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} = \frac{x + 5}{x + 10} $$

2. $$ \frac{3x + 2}{3x + 4} $$

3. $$ \frac{1}{2} $$

4. $$ \frac{x + 5}{x + 10} $$

5. $$ \frac{17x + 10}{32x + 20} = \frac{17x + 10}{4(8x + 5)} $$
The simplified forms are: 1. $$ \frac{x + 5}{x + 10} $$ 2. $$ \frac{3x + 2}{3x + 4} $$ 3. $$ \frac{1}{2} $$ 4. $$ \frac{x + 5}{x + 10} $$ 5. $$ \frac{17x + 10}{4(8x + 5)} $$

Simplify:

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