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To simplify the given expression:

\[
\frac{16x + 34}{(2x + 2)(2x + 4)}
\]

we can follow these steps:

1. **Factor the Numerator**:
   The numerator is \(16x + 34\). We can factor out 2 from this expression:
   \[
   16x + 34 = 2(8x + 17)
   \]
   So the expression becomes:
   \[
   \frac{2(8x + 17)}{(2x + 2)(2x + 4)}
   \]

2. **Factor the Denominator**:
   In the denominator, we can factor each term individually:
   \[
   (2x + 2) = 2(x + 1)
   \]
   \[
   (2x + 4) = 2(x + 2)
   \]
   Therefore, the denominator becomes:
   \[
   (2x + 2)(2x + 4) = 2(x + 1) \cdot 2(x + 2) = 4(x + 1)(x + 2)
   \]

3. **Simplify the Fraction**:
   Substitute the factored forms of the numerator and the denominator:
   \[
   \frac{2(8x + 17)}{4(x + 1)(x + 2)}
   \]
   Now, we can simplify by canceling the common factor of 2 in the numerator and the 4 in the denominator:
   \[
   \frac{2(8x + 17)}{4(x + 1)(x + 2)} = \frac{8x + 17}{2(x + 1)(x + 2)}
   \]

So the simplified form of the expression is:

\[
\frac{8x + 17}{2(x + 1)(x + 2)}
\]

Would you like a further breakdown of any step?

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Here are 5 related questions for further exploration:

1. How would you factor a different quadratic expression in the numerator?
2. What if the denominator terms had higher degrees; how would that affect simplification?
3. Can you always cancel out terms in the numerator and denominator in a rational expression?
4. How would you solve if the numerator or denominator was a polynomial of higher degree?
5. What role does factoring play in simplifying rational expressions?

**Tip:** Always factor both the numerator and denominator fully before attempting to simplify a rational expression.

Simplify the expression: (16x + 34) / ((2x + 2)(2x + 4)).

Learn how to simplify the rational expression (16x + 34) / ((2x + 2)(2x + 4)). This solution includes factoring techniques and step-by-step simplification, aimed at helping Grade 9 students understand algebraic expressions and rational simplification.

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