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Let's work through the given expression:

\[
\frac{y^2 + 4y}{y^2 - 16}
\]

### Step 1: Factor both the numerator and denominator (if possible).
1. The **numerator** is \( y^2 + 4y \). We can factor this as:
   \[
   y^2 + 4y = y(y + 4)
   \]

2. The **denominator** is \( y^2 - 16 \), which is a difference of squares. This can be factored as:
   \[
   y^2 - 16 = (y - 4)(y + 4)
   \]

### Step 2: Write the expression with the factored forms:
\[
\frac{y(y + 4)}{(y - 4)(y + 4)}
\]

### Step 3: Simplify by canceling out common factors.
Since \( (y + 4) \) appears in both the numerator and denominator, we can cancel it out (as long as \( y \neq -4 \), to avoid division by zero):
\[
\frac{y}{y - 4}
\]

### Step 4: Identify \( f(y) \) and \( g(y) \).
From the simplified expression \( \frac{y}{y - 4} \), we can write it in the form \( \frac{f(y)}{g(y)} \), where:
- \( f(y) = y \)
- \( g(y) = y - 4 \)

### Final Answer:
- \( f(y) = y \)
- \( g(y) = y - 4 \)

Would you like further details or have any questions?

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Here are 5 related questions to expand this topic:
1. How would the simplification change if the numerator had an additional constant term?
2. What happens if \( y = -4 \) in this expression?
3. How would you simplify the expression \( \frac{y^2 + 4y + 4}{y^2 - 16} \)?
4. What if the denominator was \( y^2 + 16 \) instead of \( y^2 - 16 \)?
5. How would you find the domain of the function \( \frac{y}{y-4} \)?

**Tip**: Always check for values of the variable that may make the denominator zero when simplifying rational expressions.

Simplify the expression (y^2 + 4y) / (y^2 - 16) and give your answer in the form of f(y)/g(y).

Learn how to simplify the rational expression (y^2 + 4y) / (y^2 - 16) using factorization. This step-by-step solution demonstrates simplifying algebraic fractions, factoring polynomials, and identifying values that make the expression undefined. Suitable for students in Grades 9-11.

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