To simplify the given expression:

\[
\frac{12y^2 + 20y - 16}{4y}
\]

### Step-by-step solution:

1. **Factor out the terms in the numerator** (if possible):
   The numerator is \( 12y^2 + 20y - 16 \).

   There's no common factor for all terms, so we leave it as is for now.

2. **Split the terms for division**:
   Each term in the numerator will be divided by \( 4y \):

   \[
   \frac{12y^2}{4y} + \frac{20y}{4y} - \frac{16}{4y}
   \]

3. **Simplify each fraction**:

   - \(\frac{12y^2}{4y} = 3y\)
   - \(\frac{20y}{4y} = 5\)
   - \(\frac{16}{4y} = 4/y\)

4. **Final simplified expression**:
   \[
   3y + 5 - \frac{4}{y}
   \]

Thus, the simplified expression is:

\[
3y + 5 - \frac{4}{y}
\]

---

Would you like more details on any of these steps?

Here are 5 related questions you could explore:
1. How would the expression change if the numerator was factored?
2. Can we solve \( 3y + 5 - \frac{4}{y} = 0 \)?
3. What are the restrictions on \( y \) for the simplified expression to be valid?
4. How would this expression behave as \( y \) approaches infinity?
5. How do you graph the function \( f(y) = 3y + 5 - \frac{4}{y} \)?

**Tip**: Always check if any terms can be factored or simplified before dividing!

Simplify the expression 12y^2 + 20y - 16 divided by 4y with a step-by-step solution. This problem covers algebraic concepts, including polynomial division and term-by-term simplification. Suitable for students in Grades 8-10.

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